Probability

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Configuring R

Functions from these packages will be used throughout this document:

library(conflicted) # check for conflicting function definitions
# library(printr) # inserts help-file output into markdown output
library(rmarkdown) # Convert R Markdown documents into a variety of formats.
library(pander) # format tables for markdown
library(ggplot2) # graphics
library(ggfortify) # help with graphics
library(dplyr) # manipulate data
library(tibble) # `tibble`s extend `data.frame`s
library(magrittr) # `%>%` and other additional piping tools
library(haven) # import Stata files
library(knitr) # format R output for markdown
library(tidyr) # Tools to help to create tidy data
library(plotly) # interactive graphics
library(dobson) # datasets from Dobson and Barnett 2018
library(parameters) # format model output tables for markdown
library(haven) # import Stata files
library(latex2exp) # use LaTeX in R code (for figures and tables)
library(fs) # filesystem path manipulations
library(survival) # survival analysis
library(survminer) # survival analysis graphics
library(KMsurv) # datasets from Klein and Moeschberger
library(parameters) # format model output tables for
library(webshot2) # convert interactive content to static for pdf
library(forcats) # functions for categorical variables ("factors")
library(stringr) # functions for dealing with strings
library(lubridate) # functions for dealing with dates and times
library(broom) # Summarizes key information about statistical objects in tidy tibbles
library(broom.helpers) # Provides suite of functions to work with regression model 'broom::tidy()' tibbles

Here are some R settings I use in this document:

rm(list = ls()) # delete any data that's already loaded into R

conflicts_prefer(dplyr::filter)
ggplot2::theme_set(
  ggplot2::theme_bw() + 
        # ggplot2::labs(col = "") +
    ggplot2::theme(
      legend.position = "bottom",
      text = ggplot2::element_text(size = 12, family = "serif")))

knitr::opts_chunk$set(message = FALSE)
options('digits' = 6)

panderOptions("big.mark", ",")
pander::panderOptions("table.emphasize.rownames", FALSE)
pander::panderOptions("table.split.table", Inf)
conflicts_prefer(dplyr::filter) # use the `filter()` function from dplyr() by default
legend_text_size = 9
run_graphs = TRUE

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Most of the content in this chapter should be review from UC Davis Epi 202.

1 Core properties of probabilities

1.1 Defining probabilities

Definition 1 (Probability measure) A probability measure, often denoted \(\Pr()\) or \(\text{P}()\), is a function whose domain is a \(\sigma\)-algebra of possible outcomes, \(\mathscr{S}\), and which satisfies the following properties:

1. For any statistical event \(A \in \mathscr{S}\), \(\Pr(A) \ge 0\).

2. The probability of the union of all outcomes (\(\Omega \stackrel{\text{def}}{=}\cup \mathscr{S}\)) is 1:

\[\Pr(\Omega) = 1\]

3. The probability of the union of countably many mutually disjoint events \(A_1, A_2, \ldots\) (where \(A_i \cap A_j = \emptyset\) for all \(i \neq j\)) is equal to the sum of their probabilities (countable additivity or sigma-additivity):

\[\Pr\!\left(\bigcup_{i=1}^{\infty} A_i\right) = \sum_{i=1}^{\infty} \Pr(A_i)\]

Property 3 (countable additivity) is stronger than finite additivity, which only requires

\[\Pr(A_1 \cup \cdots \cup A_n) = \sum_{i=1}^{n} \Pr(A_i)\]

for every finite collection of mutually disjoint events. Countable additivity implies finite additivity (set \(A_{n+1} = A_{n+2} = \cdots = \emptyset\) in property 3, using \(\Pr(\emptyset) = 0\)), but not vice versa: there exist set functions that satisfy finite additivity but fail countable additivity (see Wikipedia: Sigma-additive set function — An additive function which is not σ-additive). Requiring countable additivity enables results such as the continuity of probability (if \(A_1 \supseteq A_2 \supseteq \cdots\) with \(\bigcap_i A_i = \emptyset\), then \(\Pr(A_i) \to 0\)) and underpins the Theorem 4 for countable partitions.

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Theorem 1 If \(A\) and \(B\) are statistical events and \(A\subseteq B\), then \(\Pr(A \cap B) = \Pr(A)\).

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ProofProof

Proof. Left to the reader for now.

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Theorem 2 \[\Pr(A) + \Pr(\neg A) = 1\]

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ProofProof

Proof. By properties 2 and 3 of Definition 1.

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Corollary 1 \[\Pr(\neg A) = 1 - \Pr(A)\]

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ProofProof

Proof. By Theorem 2 and algebra.

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Corollary 2 If the probability of an outcome \(A\) is \(\Pr(A)=\pi\), then the probability that \(A\) does not occur is:

\[\Pr(\neg A)= 1 - \pi\]

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ProofProof

Proof. Using Corollary 1:

\[ \begin{aligned} \Pr(\neg A) &= 1 - \Pr(A) \\ &= 1 - \pi \end{aligned} \]

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1.2 Conditional probability

Definition 2 (Conditional probability) For two events \(A\) and \(B\) with \(\Pr(B) > 0\), the conditional probability of \(A\) given \(B\), denoted \(\Pr(A \mid B)\), is:

\[\Pr(A \mid B) \stackrel{\text{def}}{=}\frac{\Pr(A \cap B)}{\Pr(B)}\]

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Theorem 3 (Law of conditional probability) For any two events \(A\) and \(B\) with \(\Pr(B) > 0\):

\[\Pr(A \cap B) = \Pr(A \mid B) \cdot\Pr(B)\]

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ProofProof

Proof. Rearranging Definition 2:

\[ \begin{aligned} \Pr(A \mid B) &= \frac{\Pr(A \cap B)}{\Pr(B)} \\ \Pr(A \cap B) &= \Pr(A \mid B) \cdot\Pr(B) \end{aligned} \]

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Example 1 (Applying the law of conditional probability) Suppose 30% of adults exercise regularly (\(\Pr(E) = 0.30\)), and among adults who exercise regularly, 60% have low blood pressure (\(\Pr(L \mid E) = 0.60\)).

Then the probability that a randomly selected adult both exercises regularly and has low blood pressure is:

\[ \begin{aligned} \Pr(L \cap E) &= \Pr(L \mid E) \cdot\Pr(E) \\&= 0.60 \cdot 0.30 \\&= 0.18 \end{aligned} \]

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Theorem 4 (Law of total probability) If \(B_1, B_2, \ldots\) is a countable partition of the sample space (i.e., countably many mutually exclusive events whose union is the entire sample space), then for any event \(A\):

\[\Pr(A) = \sum_{i=1}^{\infty} \Pr(A \mid B_i) \cdot\Pr(B_i)\]

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ProofProof

Proof. Since \(B_1, B_2, \ldots\) partition the sample space, the events \(A \cap B_1, A \cap B_2, \ldots\) are mutually exclusive and their union is \(A\). By property 3 of Definition 1 (countable additivity), and then by Theorem 3:

\[ \begin{aligned} \Pr(A) &= \sum_{i=1}^{\infty} \Pr(A \cap B_i) \\&= \sum_{i=1}^{\infty} \Pr(A \mid B_i) \cdot\Pr(B_i) \end{aligned} \]

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Theorem 5 (Bayes’ theorem) For any two events \(A\) and \(B\) with \(\Pr(A) > 0\) and \(\Pr(B) > 0\):

\[\Pr(A \mid B) = \frac{\Pr(B \mid A) \cdot\Pr(A)}{\Pr(B)}\]

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ProofProof

Proof. Apply Definition 2 to both \(\Pr(A \mid B)\) and \(\Pr(B \mid A)\):

\[ \begin{aligned} \Pr(A \mid B) &= \frac{\Pr(A \cap B)}{\Pr(B)} \\&= \frac{\Pr(B \mid A) \cdot\Pr(A)}{\Pr(B)} \end{aligned} \]

The second equality follows from Theorem 3 applied to \(\Pr(B \cap A) = \Pr(B \mid A) \cdot\Pr(A)\).

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Example 2 (Positive predictive value of a medical test) Suppose a disease test has 99% sensitivity and 99% specificity, and the prevalence of the disease in the population is 7%.

Let \(D\) be the event “person has the disease” and \(+\) be the event “test is positive”. Then:

• \(\Pr(+ \mid D) = 0.99\) (sensitivity)
• \(\Pr(\neg + \mid \neg D) = 0.99\) (specificity), so the false positive rate is \(\Pr(+ \mid \neg D) = 1 - 0.99 = 0.01\)
• \(\Pr(D) = 0.07\) (prevalence)

By Bayes’ theorem (Theorem 5) and the law of total probability (Theorem 4):

\[ \begin{aligned} \Pr(D \mid +) &= \frac{\Pr(+ \mid D) \cdot\Pr(D)}{\Pr(+)} \\&= \frac{\Pr(+ \mid D) \cdot\Pr(D)}{\Pr(+ \mid D) \cdot\Pr(D) + \Pr(+ \mid \neg D) \cdot\Pr(\neg D)} \\&= \frac{0.99 \cdot 0.07}{0.99 \cdot 0.07 + 0.01 \cdot 0.93} \\&= \frac{0.0693}{0.0693 + 0.0093} \\&= \frac{0.0693}{0.0786} \\&\approx 0.88 \end{aligned} \]

Even with a highly accurate test (99% sensitive and 99% specific), only about 88% of people who test positive actually have the disease, because the disease prevalence is relatively low (7%).

2 Key probability distributions

Some distributions are typically used for outcome models (Table 1); other distributions are typically used for test statistics (Table 2).

Table 1: Distributions typically used for outcome models

Distribution	Uses
Bernoulli	Binary outcomes
Binomial	Sums of Bernoulli outcomes
Poisson	unbounded count outcomes
Geometric	Counts of non-events before an event occurs
Negative binomal	Mixtures of Poisson distributions, counts of non-events until a given number of events occurs
Normal (Gaussian)	Continuous outcomes without a more specific distribution
exponential	Time to event outcomes
Gamma	Time to event outcomes
Weibull	Time to event outcomes
Log-normal	Time to event outcomes

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Table 2: Distributions typically used for test statistics

Distribution	Uses
\(\chi^2\)	Regression comparisons (asymptotic), contingency table independence tests, goodness-of-fit tests
\(F\)	Gaussian model comparisons (exact)
\(Z\) (standard normal)	Proportions, means, regression coefficients (asymptotic)
\(T\)	Means, regression coefficients in Gaussian outcome models (exact)

2.1 The Bernoulli distribution

Definition 3 (Bernoulli distribution) The Bernoulli distribution family for a random variable \(X\) is defined as:

\[ \begin{aligned} \Pr(X=x) &= \text{1}_{x\in {\left\{0,1\right\}}}\pi^x(1-\pi)^{1-x}\\ &= \left\{{\pi, x=1}\atop{1-\pi, x=0}\right. \end{aligned} \]

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2.2 The Poisson distribution

Figure 1: “Les Poissons”

Siméon Denis Poisson

Les Poissons

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Exercise 1 Define the Poisson distribution.

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SolutionSolution

Solution 1.

Definition 4 (Poisson distribution) \[\text{P}(Y = y) = \frac{\mu^{y} e^{-\mu}}{y!}, y \in \text{N} \tag{1}\]

(see Figure 2)

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Exercise 2 What is the range of possible values for a Poisson distribution?

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SolutionSolution

Solution 2. \[\mathcal{R}(Y) = {\left\{0, 1, 2, ...\right\}} = \text{N}\]

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Theorem 6 (CDF of Poisson distribution) \[\text{P}(Y \le y) = e^{-\mu} \sum_{j=0}^{\left \lfloor{y}\right \rfloor}\frac{\mu^j}{j!} \tag{2}\]

(see Figure 3)

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library(dplyr)
pois_dists <- tibble(
  mu = c(0.5, 1, 2, 5, 10, 20)
) |>
  reframe(
    .by = mu,
    x = 0:30
  ) |>
  mutate(
    `P(X = x)` = dpois(x, lambda = mu),
    `P(X <= x)` = ppois(x, lambda = mu),
    mu = factor(mu)
  )

library(ggplot2)
library(latex2exp)

plot0 <- pois_dists |>
  ggplot(
    aes(
      x = x,
      y = `P(X = x)`,
      fill = mu,
      col = mu
    )
  ) +
  theme(legend.position = "bottom") +
  labs(
    fill = latex2exp::TeX("$\\mu$"),
    col = latex2exp::TeX("$\\mu$"),
    y = latex2exp::TeX("$\\Pr_{\\mu}(X = x)$")
  )

plot1 <- plot0 +
  geom_segment(yend = 0) +
  facet_wrap(~mu)

print(plot1)

Figure 2: Poisson PMFs, by mean parameter \(\mu\)

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library(ggplot2)

plot2 <-
  plot0 +
  geom_step(alpha = 0.75) +
  aes(y = `P(X <= x)`) +
  labs(y = latex2exp::TeX("$\\Pr_{\\mu}(X \\leq x)$"))

print(plot2)

Figure 3: Poisson CDFs

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Exercise 3 (Poisson distribution functions) Let \(X \sim \text{Pois}(\mu = 3.75)\).

Compute:

• \(\text{P}(X = 4 | \mu = 3.75)\)
• \(\text{P}(X \le 7 | \mu = 3.75)\)
• \(\text{P}(X > 5 | \mu = 3.75)\)

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SolutionSolution

Solution.

• \(\text{P}(X=4) = 0.19378\)
• \(\text{P}(X\le 7) = 0.962379\)
• \(\text{P}(X > 5) = 0.177117\)

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Theorem 7 (Properties of the Poisson distribution) If \(X \sim \text{Pois}(\mu)\), then:

• \(\text{E}[X] = \mu\)
• \(\text{Var}(X) = \mu\)
• \(\text{P}(X=x) = \frac{\mu}{x} \text{P}(X = x-1)\)
• For \(x < \mu\), \(\text{P}(X=x) > \text{P}(X = x-1)\)
• For \(x = \mu\), \(\text{P}(X=x) = \text{P}(X = x-1)\)
• For \(x > \mu\), \(\text{P}(X=x) < \text{P}(X = x-1)\)
• \(\arg \max_{x} \text{P}(X=x) = \left \lfloor{\mu}\right \rfloor\)

Exercise 4 Prove Theorem 7.

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SolutionSolution

Solution. \[ \begin{aligned} \text{E}[X] &= \sum_{x=0}^\infty x \cdot P(X=x)\\ &= 0 \cdot P(X=0) + \sum_{x=1}^\infty x \cdot P(X=x)\\ &= 0 + \sum_{x=1}^\infty x \cdot P(X=x)\\ &= \sum_{x=1}^\infty x \cdot P(X=x)\\ &= \sum_{x=1}^\infty x \cdot \frac{\lambda^x e^{-\lambda}}{x!}\\ &= \sum_{x=1}^\infty x \cdot \frac{\lambda^x e^{-\lambda}}{x \cdot (x-1)!} & [\text{definition of factorial ("!") function}]\\ &= \sum_{x=1}^\infty \frac{\lambda^x e^{-\lambda}}{ (x-1)!}\\ &= \sum_{x=1}^\infty \frac{(\lambda \cdot \lambda^{x-1}) e^{-\lambda}}{ (x-1)!}\\ &= \lambda \cdot \sum_{x=1}^\infty \frac{( \lambda^{x-1}) e^{-\lambda}}{ (x-1)!}\\ &= \lambda \cdot \sum_{y=0}^\infty \frac{( \lambda^{y}) e^{-\lambda}}{ (y)!} &[\text{substituting } y \stackrel{\text{def}}{=}x-1]\\ &= \lambda \cdot 1 &[\text{because PDFs sum to 1}]\\ &= \lambda\\ \end{aligned} \]

See also https://statproofbook.github.io/P/poiss-mean.

For the variance, see https://statproofbook.github.io/P/poiss-var.

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Accounting for exposure

Definition 5 (Exposure magnitude) For many count outcomes, there is some sense of an exposure magnitude, such as population size, or duration of observation, which multiplicatively rescales the expected (mean) count.

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Exercise 5 What are some examples of exposure magnitudes?

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SolutionSolution

Solution.

Table 3: Examples of exposure units

outcome	exposure units
disease incidence	number of individuals exposed; time at risk
car accidents	miles driven
worksite accidents	person-hours worked
population size	size of habitat

Exposure units are similar to the number of trials in a binomial distribution, but in non-binomial count outcomes, there can be more than one event per unit of exposure.

We can use \(t\) to represent continuous-valued exposures/observation durations, and \(n\) to represent discrete-valued exposures.

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Definition 6 (Event rate)  

For a count outcome \(Y\) with exposure magnitude \(t\), the event rate (denoted \(\lambda\)) is defined as the mean of \(Y\) divided by the exposure magnitude. That is:

\[\mu \stackrel{\text{def}}{=}\text{E}[Y|T=t]\]

\[\lambda \stackrel{\text{def}}{=}\frac{\mu}{t} \tag{3}\]

Event rate is somewhat analogous to odds in binary outcome models; it typically serves as an intermediate transformation between the mean of the outcome and the linear component of the model. However, in contrast with the odds function, the transformation \(\lambda = \mu/t\) is not considered part of the Poisson model’s link function, and it treats the exposure magnitude covariate differently from the other covariates.

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Theorem 8 (Transformation function from event rate to mean) For a count variable with mean \(\mu\), event rate \(\lambda\), and exposure magnitude \(t\):

\[\mu = \lambda \cdot t \tag{4}\]

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SolutionSolution

Solution. Start from definition of event rate and use algebra to solve for \(\mu\).

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Equation 4 is analogous to the inverse-odds function for binary variables.

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Theorem 9 When the exposure magnitude is 0, there is no opportunity for events to occur:

\[\text{E}[Y|T=0] = 0\]

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ProofProof

Proof. \[\text{E}[Y|T=0] = \lambda \cdot 0 = 0\]

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Important

The exposure magnitude, \(T\), is similar to a covariate in linear or logistic regression. However, there is an important difference: in count regression, there is no intercept corresponding to \(\text{E}[Y|T=0]\). In other words, this model assumes that if there is no exposure, there can’t be any events.

Theorem 10 If \(\mu = \lambda\cdot t\), then:

\[\log{\mu} = \log{\lambda} + \log{t}\]

Definition 7 (Offset) When the linear component of a model involves a term without an unknown coefficient, that term is called an offset.

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Theorem 11 If \(X\) and \(Y\) are independent Poisson random variables with means \(\mu_X\) and \(\mu_Y\), their sum, \(Z=X+Y\), is also a Poisson random variable, with mean \(\mu_Z = \mu_X + \mu_Y\).

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ProofProof

Proof. See https://web.stanford.edu/class/archive/cs/cs109/cs109.1206/lectureNotes/LN12_independent_rvs.pdf, Example 3.

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2.3 The Negative-Binomial distribution

Definition 8 (Negative binomial distribution) \[ \text{P}(Y=y) = \frac{\mu^y}{y!} \cdot \frac{\Gamma(\rho + y)}{\Gamma(\rho) \cdot (\rho + \mu)^y} \cdot \left(1+\frac{\mu}{\rho}\right)^{-\rho} \]

where \(\rho\) is an overdispersion parameter and \(\Gamma(x) = (x-1)!\) for integers \(x\).

You don’t need to memorize or understand this expression.

As \(\rho \rightarrow \infty\), the second term converges to 1 and the third term converges to \(\text{exp}{\left\{-\mu\right\}}\), which brings us back to the Poisson distribution.

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Theorem 12 If \(Y \sim \text{NegBin}(\mu, \rho)\), then:

• \(\text{E}[Y] = \mu\)
• \(\text{Var}{\left(Y\right)} = \mu + \frac{\mu^2}{\rho} > \mu\)

2.4 Weibull Distribution

\[ \begin{aligned} p(t)&= \alpha\lambda x^{\alpha-1}\text{e}^{-\lambda x^\alpha}\\ {\lambda}(t)&=\alpha\lambda x^{\alpha-1}\\ \text{S}(t)&=\text{e}^{-\lambda x^\alpha}\\ E(T)&= \Gamma(1+1/\alpha)\cdot \lambda^{-1/\alpha} \end{aligned} \]

When \(\alpha=1\) this is the exponential. When \(\alpha>1\) the hazard is increasing and when \(\alpha < 1\) the hazard is decreasing. This provides more flexibility than the exponential.

We will see more of this distribution later.

3 Characteristics of probability distributions

3.1 Probability density function

Definition 9 (probability density) If \(X\) is a continuous random variable, then the probability density of \(X\) at value \(x\), denoted \(f(x)\), \(f_X(x)\), \(\text{p}(x)\), \(\text{p}_X(x)\), or \(\text{p}(X=x)\), is defined as the limit of the probability (mass) that \(X\) is in an interval around \(x\), divided by the width of that interval, as that width reduces to 0.

\[ \begin{aligned} f(x) &\stackrel{\text{def}}{=}\lim_{\Delta \rightarrow 0} \frac{\text{P}(X \in [x, x + \Delta])}{\Delta} \end{aligned} \]

See also Rothman et al. (2021) (Chapter 22, p. 535) and https://en.wikipedia.org/wiki/Probability_density_function#Formal_definition

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Definition 10 (Cumulative distribution function (CDF)) For a random variable \(X\), its population CDF is \[F(t)=\Pr(X\le t), \quad t\in\mathbb{R}.\]

Definition 11 (Quantile function (population inverse CDF)) For a random variable \(X\) with cumulative distribution function (CDF) \(F\), its population quantile function (generalized inverse of \(F\)) is \[Q(p)=\inf\{t:F(t)\ge p\}, \quad 0<p\le 1.\]

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Theorem 13 (Density function is derivative of CDF) The density function \(f(t)\) or \(\text{p}(T=t)\) for a random variable \(T\) at value \(t\) is equal to the derivative of the cumulative probability function \(F(t) \stackrel{\text{def}}{=}P(T\le t)\); that is:

\[f(t) \stackrel{\text{def}}{=}\frac{\partial}{\partial t} F(t)\]

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Theorem 14 (Density functions integrate to 1) For any density function \(f(x)\),

\[\int_{x \in \mathcal{R}(X)} f(x) dx = 1\]

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3.2 Hazard function

Definition 12 (Hazard function, hazard rate, hazard rate function)  

The hazard function, hazard rate, hazard rate function, for a random variable \(T\) at value \(t\), typically denoted as \(\text{h}(t)\) ¹ or \(\lambda(t)\), ² is the conditional density of \(T\) at \(t\), given \(T \ge t\). That is:

\[{\lambda}(t) \stackrel{\text{def}}{=}\text{p}(T=t|T\ge t)\]

If \(T\) represents the time at which an event occurs, then \({\lambda}(t)\) is the probability that the event occurs at time \(t\), given that it has not occurred prior to time \(t\).

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Table 4: Probability distribution functions

Name	Symbols	Definition
Probability density function (PDF)	\(\text{f}(t), \text{p}(t)\)	\(\text{p}(T=t)\)
Cumulative distribution function (CDF)	\(\text{F}(t), \text{P}(t)\)	\(\text{P}(T\leq t)\)
Survival function	\(\text{S}(t), \bar{\text{F}}(t)\)	\(\text{P}(T > t)\)
Hazard function	\(\lambda(t), \text{h}(t)\)	\(\text{p}(T=t|T\ge t)\)
Cumulative hazard function	\(\Lambda(t), \text{H}(t)\)	\(\int_{u=-\infty}^t {\lambda}(u)du\)
Log-hazard function	\(\eta(t)\)	\(\text{log}{\left\{{\lambda}(t)\right\}}\)

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\[ \text{f}(t) \xleftarrow[\text{S}(t){\lambda}(t)]{-S'(t)} \text{S}(t) \xleftarrow[]{\text{exp}{\left\{-{\Lambda}(t)\right\}}} {\Lambda}(t) \xleftarrow[]{\int_{u=0}^t {\lambda}(u)du} {\lambda}(t) \xleftarrow[]{\text{exp}{\left\{\eta(t)\right\}}} \eta(t) \]

\[ \text{f}(t) \xrightarrow[\int_{u=t}^\infty \text{f}(u)du]{\text{f}(t)/{\lambda}(t)} \text{S}(t) \xrightarrow[-\log{\text{S}(t)}]{} {\Lambda}(t) \xrightarrow[{\Lambda}'(t)]{} {\lambda}(t) \xrightarrow[\text{log}{\left\{{\lambda}(t)\right\}}]{} \eta(t) \]

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3.3 Expectation

Definition 13 (Expectation, expected value, population mean ) The expectation, expected value, or population mean of a continuous random variable \(X\), denoted \(\text{E}{\left[X\right]}\), \(\mu(X)\), or \(\mu_X\), is the weighted mean of \(X\)’s possible values, weighted by the probability density function of those values:

\[\text{E}{\left[X\right]} = \int_{x\in \mathcal{R}(X)} x \cdot \text{p}(X=x)dx\]

The expectation, expected value, or population mean of a discrete random variable \(X\), denoted \(\text{E}{\left[X\right]}\), \(\mu(X)\), or \(\mu_X\), is the mean of \(X\)’s possible values, weighted by the probability mass function of those values:

\[\text{E}{\left[X\right]} = \sum_{x \in \mathcal{R}(X)} x \cdot \text{P}(X=x)\]

(c.f. https://en.wikipedia.org/wiki/Expected_value)

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Theorem 15 (Expectation of the Bernoulli distribution) The expectation of a Bernoulli random variable with parameter \(\pi\) is:

\[\text{E}{\left[X\right]} = \pi\]

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ProofProof

Proof. \[ \begin{aligned} \text{E}{\left[X\right]} &= \sum_{x\in \mathcal{R}(X)} x \cdot\text{P}(X=x) \\&= \sum_{x\in {\left\{0,1\right\}}} x \cdot\text{P}(X=x) \\&= {\left(0 \cdot\text{P}(X=0)\right)} + {\left(1 \cdot\text{P}(X=1)\right)} \\&= {\left(0 \cdot(1-\pi)\right)} + {\left(1 \cdot\pi\right)} \\&= 0 + \pi \\&= \pi \end{aligned} \]

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Theorem 16 (Expectation of time-to-event variables) If \(T\) is a non-negative random variable, then:

\[\mu(T|\tilde{X}= \tilde{x}) = \int_{t=0}^{\infty}\text{S}(t)dt\]

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Theorem 17 (Law of the Unconscious Statistician (LOTUS)) For any function \(g\) of a discrete random variable \(X\):

\[\text{E}{\left[g(X)\right]} = \sum_{x \in \mathcal{R}(X)} g(x) \cdot\text{P}(X=x)\]

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ProofProof

Proof. Let \(Y = g(X)\). By Definition 13 applied to \(Y\):

\[ \begin{aligned} \text{E}{\left[g(X)\right]} &= \text{E}{\left[Y\right]} \\&= \sum_{y \in \mathcal{R}(Y)} y \cdot\text{P}(Y=y) \\&= \sum_{y \in \mathcal{R}(Y)} y \cdot\text{P}(g(X)=y) \\&= \sum_{y \in \mathcal{R}(Y)} y \cdot\sum_{\substack{x \in \mathcal{R}(X) \\ g(x) = y}} \text{P}(X=x) \\&= \sum_{x \in \mathcal{R}(X)} g(x) \cdot\text{P}(X=x) \end{aligned} \]

where the last equality follows by rearranging the double sum, grouping each term \(x\) by its image \(y = g(x)\).

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LOTUS says that to compute \(\text{E}{\left[g(X)\right]}\), we do not need to first find the distribution of \(g(X)\); we can compute the expectation directly using the distribution of \(X\).

For a continuous random variable \(X\) with density \(\text{p}(X=x)\), the analogous formula is:

\[\text{E}{\left[g(X)\right]} = \int_{x \in \mathcal{R}(X)} g(x) \cdot\text{p}(X=x)\, dx\]

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Example 3 (Expected value of \(X^2\) for a Bernoulli variable) Let \(X \sim \text{Ber}(\pi)\). By LOTUS (Theorem 17):

\[ \begin{aligned} \text{E}{\left[X^2\right]} &= \sum_{x \in {\left\{0,1\right\}}} x^2 \cdot\text{P}(X=x) \\&= 0^2 \cdot\text{P}(X=0) + 1^2 \cdot\text{P}(X=1) \\&= 0^2 \cdot(1-\pi) + 1^2 \cdot\pi \\&= 0 + \pi \\&= \pi \end{aligned} \]

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Definition 14 (Conditional expectation) Discrete case. Let \(X\) and \(Y\) be jointly distributed discrete random variables. The conditional probability mass function of \(Y\) given \(X = x\) (for values of \(x\) with \(\text{P}(X = x) > 0\)) is:

\[\text{P}(Y = y \mid X = x) \stackrel{\text{def}}{=}\frac{\text{P}(X = x,\, Y = y)}{\text{P}(X = x)}\]

The conditional expectation of \(Y\) given \(X = x\) is:

\[\text{E}{\left[Y \mid X = x\right]} \stackrel{\text{def}}{=}\sum_{y \in \mathcal{R}(Y)} y \cdot\text{P}(Y = y \mid X = x)\]

Continuous case. Let \(X\) and \(Y\) be jointly distributed continuous random variables with joint density \(\text{p}(X = x,\, Y = y)\) and marginal density \(\text{p}(X = x)\). The conditional probability density function of \(Y\) given \(X = x\) (for values of \(x\) with \(\text{p}(X = x) > 0\)) is:

\[\text{p}(Y = y \mid X = x) \stackrel{\text{def}}{=}\frac{\text{p}(X = x,\, Y = y)}{\text{p}(X = x)}\]

The conditional expectation of \(Y\) given \(X = x\) is:

\[\text{E}{\left[Y \mid X = x\right]} \stackrel{\text{def}}{=}\int_{y \in \mathcal{R}(Y)} y \cdot\text{p}(Y = y \mid X = x)\, dy\]

Conditional expectation function. The conditional expectation function \(\text{E}{\left[Y \mid X\right]}\) is the function (and hence random variable) of \(X\) obtained by evaluating \(\text{E}{\left[Y \mid X = x\right]}\) at \(X\); that is, \(\text{E}{\left[Y \mid X\right]} = g(X)\) where \(g(x) \stackrel{\text{def}}{=}\text{E}{\left[Y \mid X = x\right]}\).

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Theorem 18 (Law of iterated expectations) For any two random variables \(X\) and \(Y\):

\[\text{E}{\left[Y\right]} = \text{E}{\left[\text{E}{\left[Y \mid X\right]}\right]}\]

Alternate names for this identity include: the tower rule, the tower property, the law of total expectation, and the smoothing theorem.

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ProofProof

Proof. Discrete case. When \(X\) and \(Y\) are discrete, applying Definition 13 to \(\text{E}{\left[\text{E}{\left[Y \mid X\right]}\right]}\) and then the law of total probability (Theorem 4) applied to the countable partition \(\{X = x : x \in \mathcal{R}(X)\}\):

\[ \begin{aligned} \text{E}{\left[\text{E}{\left[Y \mid X\right]}\right]} &= \sum_{x \in \mathcal{R}(X)} \text{E}{\left[Y \mid X=x\right]} \cdot\text{P}(X=x) \\&= \sum_{x \in \mathcal{R}(X)} {\left(\sum_{y \in \mathcal{R}(Y)} y \cdot\text{P}(Y=y \mid X=x)\right)} \cdot\text{P}(X=x) \\&= \sum_{y \in \mathcal{R}(Y)} y \cdot\sum_{x \in \mathcal{R}(X)} \text{P}(Y=y \mid X=x) \cdot\text{P}(X=x) \\&= \sum_{y \in \mathcal{R}(Y)} y \cdot\text{P}(Y=y) \\&= \text{E}{\left[Y\right]} \end{aligned} \]

Continuous case. When \(X\) and \(Y\) are continuous, applying Definition 13 to \(\text{E}{\left[\text{E}{\left[Y \mid X\right]}\right]}\) and then using Definition 14 for \(\text{E}{\left[Y \mid X=x\right]}\):

\[ \begin{aligned} \text{E}{\left[\text{E}{\left[Y \mid X\right]}\right]} &= \int_{x \in \mathcal{R}(X)} \text{E}{\left[Y \mid X=x\right]} \cdot\text{p}(X=x)\, dx \\&= \int_{x \in \mathcal{R}(X)} {\left(\int_{y \in \mathcal{R}(Y)} y \cdot\text{p}(Y=y \mid X=x)\, dy\right)} \cdot\text{p}(X=x)\, dx \\&= \int_{y \in \mathcal{R}(Y)} y \cdot{\left(\int_{x \in \mathcal{R}(X)} \text{p}(Y=y \mid X=x) \cdot\text{p}(X=x)\, dx\right)}\, dy \\&= \int_{y \in \mathcal{R}(Y)} y \cdot\text{p}(Y=y)\, dy \\&= \text{E}{\left[Y\right]} \end{aligned} \]

where the third equality exchanges the order of integration by Fubini’s theorem, and the fourth equality uses \(\int_{x} \text{p}(Y=y \mid X=x) \cdot\text{p}(X=x)\, dx = \int_{x} \text{p}(X=x, Y=y)\, dx = \text{p}(Y=y)\) (marginalization of the joint density).

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Theorem 19 (Conditional law of iterated expectations) For random variables \(X\), \(Y\), and \(Z\):

\[\text{E}{\left[Y \mid Z\right]} = \text{E}{\left[\text{E}{\left[Y \mid X,Z\right]} \mid Z\right]}\]

This is the tower rule applied conditionally on \(Z\).

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ProofProof

Proof. For each fixed value \(z\) with positive probability or density:

Discrete case. Conditioning on \(Z=z\), and applying the law of total probability to the partition \(\{X=x : x \in \mathcal{R}(X)\}\) under the conditional distribution given \(Z=z\):

\[ \begin{aligned} \text{E}{\left[\text{E}{\left[Y \mid X,Z\right]} \mid Z=z\right]} &= \sum_{x \in \mathcal{R}(X)} \text{E}{\left[Y \mid X=x,Z=z\right]} \cdot\text{P}(X=x \mid Z=z) \\&= \text{E}{\left[Y \mid Z=z\right]} \end{aligned} \]

Continuous case. Conditioning on \(Z=z\), and integrating over \(X\) under the conditional density \(\text{p}(X=x \mid Z=z)\):

\[ \begin{aligned} \text{E}{\left[\text{E}{\left[Y \mid X,Z\right]} \mid Z=z\right]} &= \int_{x \in \mathcal{R}(X)} \text{E}{\left[Y \mid X=x,Z=z\right]} \cdot\text{p}(X=x \mid Z=z)\, dx \\&= \text{E}{\left[Y \mid Z=z\right]} \end{aligned} \]

Therefore, as random variables of \(Z\), \(\text{E}{\left[Y \mid Z\right]} = \text{E}{\left[\text{E}{\left[Y \mid X,Z\right]} \mid Z\right]}\).

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Example 4 (Marginal expectation from conditional expectations) Suppose \(X\) is a binary random variable indicating treatment assignment (\(X=1\) treated, \(X=0\) control), with \(\text{P}(X=1) = 0.5\), and suppose the outcome \(Y\) has conditional expectations:

\[\text{E}{\left[Y \mid X=1\right]} = 10, \quad \text{E}{\left[Y \mid X=0\right]} = 6\]

By the law of iterated expectations (Theorem 18):

\[ \begin{aligned} \text{E}{\left[Y\right]} &= \text{E}{\left[\text{E}{\left[Y \mid X\right]}\right]} \\&= \text{E}{\left[Y \mid X=1\right]} \cdot\text{P}(X=1) + \text{E}{\left[Y \mid X=0\right]} \cdot\text{P}(X=0) \\&= 10 \cdot 0.5 + 6 \cdot 0.5 \\&= 5 + 3 \\&= 8 \end{aligned} \]

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Definition 15 (Expectation of a random matrix) For a random matrix \(\mathbf{A}\) of size \(m \times n\) with \((i,j)\)-th element \(A_{ij}\), the expectation \(\text{E}\mathbf{A}\) is the \(m \times n\) matrix whose \((i,j)\)-th element is \(\text{E}{\left[A_{ij}\right]}\):

\[ \text{E}\mathbf{A} \stackrel{\text{def}}{=}\begin{pmatrix} \text{E}{\left[A_{11}\right]} & \text{E}{\left[A_{12}\right]} & \cdots & \text{E}{\left[A_{1n}\right]} \\ \text{E}{\left[A_{21}\right]} & \text{E}{\left[A_{22}\right]} & \cdots & \text{E}{\left[A_{2n}\right]} \\ \vdots & \vdots & \ddots & \vdots \\ \text{E}{\left[A_{m1}\right]} & \text{E}{\left[A_{m2}\right]} & \cdots & \text{E}{\left[A_{mn}\right]} \end{pmatrix} \]

In other words, expectation is applied element-wise to a random matrix.

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3.4 Deviation, error, and noise

Definition 16 (Deviation) A deviation is the difference between a value and a reference value. For any quantity \(z\) and reference value \(r\):

\[z - r\]

In probability and statistics, “deviation” often means deviation from a population mean. For a random variable \(Y\):

\[Y - \text{E}{\left[Y\right]}\]

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Definition 17 (Deviation from a population or subpopulation mean) In probabilistic models, we call this quantity a deviation from a mean. It is often also called an error or noise term in other sources. For the random variable \(Y\), define the deviation from its mean as:

\[e(Y) \stackrel{\text{def}}{=}Y - \text{E}{\left[Y\right]}\]

For a realized observation \(y\): \[e(y) \stackrel{\text{def}}{=}y - \text{E}{\left[Y\right]}\]

In regression settings, the reference mean is often conditional on covariates: \(e(y_i) \stackrel{\text{def}}{=}y_i - \text{E}{\left[Y_i \mid X_i\right]}\).

In this course, we prefer “deviation” for this mean-deviation quantity. The terms “error” and “noise” are common aliases. We use “residual” (defined in the Linear regression chapter) for deviations from fitted values. For notation in this course, we use \(e(\cdot)\) for these model/data deviations, and reserve \(\varepsilon{\left(\cdot\right)}\) for estimator-to-estimand deviations (see Estimation).

See:

• Wikipedia: Errors and residuals
• Wikipedia: Deviation (statistics)
• Wikipedia: Linear regression — Notation and terminology

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3.5 Variance and related characteristics

Definition 18 (Variance) The variance of a random variable \(X\) is the expectation of the squared difference between \(X\) and \(\text{E}{\left[X\right]}\); that is:

\[ \text{Var}{\left(X\right)} \stackrel{\text{def}}{=}\text{E}{\left[(X-\text{E}{\left[X\right]})^2\right]} \]

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Theorem 20 (Simplified expression for variance) \[\text{Var}{\left(X\right)}=\text{E}{\left[X^2\right]} - {\left(\text{E}{\left[X\right]}\right)}^2\]

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ProofProof

Proof. By linearity of expectation, we have:

\[ \begin{aligned} \text{Var}{\left(X\right)} &\stackrel{\text{def}}{=}\text{E}{\left[(X-\text{E}{\left[X\right]})^2\right]}\\ &=\text{E}{\left[X^2 - 2X\text{E}{\left[X\right]} + {\left(\text{E}{\left[X\right]}\right)}^2\right]}\\ &=\text{E}{\left[X^2\right]} - \text{E}{\left[2X\text{E}{\left[X\right]}\right]} + \text{E}{\left[{\left(\text{E}{\left[X\right]}\right)}^2\right]}\\ &=\text{E}{\left[X^2\right]} - 2\text{E}{\left[X\right]}\text{E}{\left[X\right]} + {\left(\text{E}{\left[X\right]}\right)}^2\\ &=\text{E}{\left[X^2\right]} - {\left(\text{E}{\left[X\right]}\right)}^2\\ \end{aligned} \]

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Theorem 21 (Law of total variance) For random variables \(X\) and \(Y\):

\[\text{Var}{\left(Y\right)} = \text{E}{\left[\text{Var}{\left(Y \mid X\right)}\right]} + \text{Var}{\left(\text{E}{\left[Y \mid X\right]}\right)}\]

where \(\text{Var}{\left(Y \mid X\right)} \stackrel{\text{def}}{=}\text{E}{\left[(Y-\text{E}{\left[Y \mid X\right]})^2 \mid X\right]}\).

Alternate names include: the conditional variance formula, Eve’s law, and the variance decomposition formula.

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ProofProof

Proof. Write \(Y-\text{E}{\left[Y\right]} = {\left(Y-\text{E}{\left[Y \mid X\right]}\right)} + {\left(\text{E}{\left[Y \mid X\right]}-\text{E}{\left[Y\right]}\right)}\). Then:

\[ \begin{aligned} {\left(Y-\text{E}{\left[Y\right]}\right)}^2 &= {\left(Y-\text{E}{\left[Y \mid X\right]}\right)}^2 + {\left(\text{E}{\left[Y \mid X\right]}-\text{E}{\left[Y\right]}\right)}^2 + 2{\left(Y-\text{E}{\left[Y \mid X\right]}\right)}{\left(\text{E}{\left[Y \mid X\right]}-\text{E}{\left[Y\right]}\right)} \end{aligned} \]

Taking expectation:

\[ \begin{aligned} \text{Var}{\left(Y\right)} &= \text{E}{\left[{\left(Y-\text{E}{\left[Y \mid X\right]}\right)}^2\right]} + \text{E}{\left[{\left(\text{E}{\left[Y \mid X\right]}-\text{E}{\left[Y\right]}\right)}^2\right]} \\&\quad + 2\text{E}{\left[{\left(Y-\text{E}{\left[Y \mid X\right]}\right)}{\left(\text{E}{\left[Y \mid X\right]}-\text{E}{\left[Y\right]}\right)}\right]} \end{aligned} \]

For the cross-term:

Discrete case.

\[ \begin{aligned} \text{E}{\left[{\left(Y-\text{E}{\left[Y \mid X\right]}\right)}{\left(\text{E}{\left[Y \mid X\right]}-\text{E}{\left[Y\right]}\right)}\right]} &= \sum_{x \in \mathcal{R}(X)} \text{E}{\left[ {\left(Y-\text{E}{\left[Y \mid X\right]}\right)} {\left(\text{E}{\left[Y \mid X\right]}-\text{E}{\left[Y\right]}\right)} \mid X=x \right]} \cdot\text{P}(X=x) \\&= \sum_{x \in \mathcal{R}(X)} {\left(\text{E}{\left[Y \mid X=x\right]}-\text{E}{\left[Y\right]}\right)} \cdot\text{E}{\left[Y-\text{E}{\left[Y \mid X=x\right]}\mid X=x\right]} \cdot\text{P}(X=x) \\&= 0 \end{aligned} \]

Continuous case.

\[ \begin{aligned} \text{E}{\left[{\left(Y-\text{E}{\left[Y \mid X\right]}\right)}{\left(\text{E}{\left[Y \mid X\right]}-\text{E}{\left[Y\right]}\right)}\right]} &= \int_{x \in \mathcal{R}(X)} \text{E}{\left[ {\left(Y-\text{E}{\left[Y \mid X\right]}\right)} {\left(\text{E}{\left[Y \mid X\right]}-\text{E}{\left[Y\right]}\right)} \mid X=x \right]} \cdot\text{p}(X=x)\, dx \\&= \int_{x \in \mathcal{R}(X)} {\left(\text{E}{\left[Y \mid X=x\right]}-\text{E}{\left[Y\right]}\right)} \cdot\text{E}{\left[Y-\text{E}{\left[Y \mid X=x\right]}\mid X=x\right]} \cdot\text{p}(X=x)\, dx \\&= 0 \end{aligned} \]

Therefore:

\[ \begin{aligned} \text{Var}{\left(Y\right)} &= \text{E}{\left[{\left(Y-\text{E}{\left[Y \mid X\right]}\right)}^2\right]} + \text{E}{\left[{\left(\text{E}{\left[Y \mid X\right]}-\text{E}{\left[Y\right]}\right)}^2\right]} \\&= \text{E}{\left[\text{Var}{\left(Y \mid X\right)}\right]} + \text{Var}{\left(\text{E}{\left[Y \mid X\right]}\right)} \end{aligned} \]

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Definition 19 (Precision) The precision of a random variable \(X\), often denoted \(\tau(X)\), \(\tau_X\), or shorthanded as \(\tau\), is the inverse of that random variable’s variance; that is:

\[\tau(X) \stackrel{\text{def}}{=}{\left(\text{Var}{\left(X\right)}\right)}^{-1}\]

Definition 20 (Standard deviation) The standard deviation of a random variable \(X\) is the square-root of the variance of \(X\):

\[\text{SD}{\left(X\right)} \stackrel{\text{def}}{=}\sqrt{\text{Var}{\left(X\right)}}\]

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Definition 21 (Covariance) For any two one-dimensional random variables, \(X,Y\):

\[\text{Cov}{\left(X,Y\right)} \stackrel{\text{def}}{=}\text{E}{\left[(X - \text{E}{\left[X\right]})(Y - \text{E}{\left[Y\right]})\right]}\]

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Theorem 22 \[\text{Cov}{\left(X,Y\right)}= \text{E}{\left[XY\right]} - \text{E}{\left[X\right]} \text{E}{\left[Y\right]}\]

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Theorem 23 (Law of total covariance) For random variables \(X\), \(Y\), and \(Z\):

\[\text{Cov}{\left(Y,Z\right)} = \text{E}{\left[\text{Cov}{\left(Y,Z \mid X\right)}\right]} + \text{Cov}{\left(\text{E}{\left[Y \mid X\right]}, \text{E}{\left[Z \mid X\right]}\right)}\]

where \(\text{Cov}{\left(Y,Z \mid X\right)} \stackrel{\text{def}}{=}\text{E}{\left[(Y-\text{E}{\left[Y \mid X\right]})(Z-\text{E}{\left[Z \mid X\right]}) \mid X\right]}\).

Alternate names include: the covariance decomposition formula and the conditional covariance formula.

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ProofProof

Proof. Write:

\[ \begin{aligned} Y-\text{E}{\left[Y\right]} &= {\left(Y-\text{E}{\left[Y \mid X\right]}\right)} + {\left(\text{E}{\left[Y \mid X\right]}-\text{E}{\left[Y\right]}\right)} \\ Z-\text{E}{\left[Z\right]} &= {\left(Z-\text{E}{\left[Z \mid X\right]}\right)} + {\left(\text{E}{\left[Z \mid X\right]}-\text{E}{\left[Z\right]}\right)} \end{aligned} \]

Then:

\[ \begin{aligned} \text{Cov}{\left(Y,Z\right)} &= \text{E}{\left[{\left(Y-\text{E}{\left[Y\right]}\right)}{\left(Z-\text{E}{\left[Z\right]}\right)}\right]} \\&= \text{E}{\left[{\left(Y-\text{E}{\left[Y \mid X\right]}\right)}{\left(Z-\text{E}{\left[Z \mid X\right]}\right)}\right]} \\&\quad + \text{E}{\left[{\left(Y-\text{E}{\left[Y \mid X\right]}\right)}{\left(\text{E}{\left[Z \mid X\right]}-\text{E}{\left[Z\right]}\right)}\right]} \\&\quad + \text{E}{\left[{\left(\text{E}{\left[Y \mid X\right]}-\text{E}{\left[Y\right]}\right)}{\left(Z-\text{E}{\left[Z \mid X\right]}\right)}\right]} \\&\quad + \text{E}{\left[{\left(\text{E}{\left[Y \mid X\right]}-\text{E}{\left[Y\right]}\right)}{\left(\text{E}{\left[Z \mid X\right]}-\text{E}{\left[Z\right]}\right)}\right]} \end{aligned} \]

For the two mixed terms:

Discrete case.

\[ \begin{aligned} \text{E}{\left[{\left(Y-\text{E}{\left[Y \mid X\right]}\right)}{\left(\text{E}{\left[Z \mid X\right]}-\text{E}{\left[Z\right]}\right)}\right]} &= \sum_{x \in \mathcal{R}(X)} \text{E}{\left[ {\left(Y-\text{E}{\left[Y \mid X\right]}\right)} {\left(\text{E}{\left[Z \mid X\right]}-\text{E}{\left[Z\right]}\right)} \mid X=x \right]} \cdot\text{P}(X=x) \\&= \sum_{x \in \mathcal{R}(X)} {\left(\text{E}{\left[Z \mid X=x\right]}-\text{E}{\left[Z\right]}\right)} \cdot\text{E}{\left[Y-\text{E}{\left[Y \mid X=x\right]} \mid X=x\right]} \cdot\text{P}(X=x) \\&= 0 \end{aligned} \]

and similarly:

\[ \text{E}{\left[{\left(\text{E}{\left[Y \mid X\right]}-\text{E}{\left[Y\right]}\right)}{\left(Z-\text{E}{\left[Z \mid X\right]}\right)}\right]}=0. \]

Continuous case.

\[ \begin{aligned} \text{E}{\left[{\left(Y-\text{E}{\left[Y \mid X\right]}\right)}{\left(\text{E}{\left[Z \mid X\right]}-\text{E}{\left[Z\right]}\right)}\right]} &= \int_{x \in \mathcal{R}(X)} \text{E}{\left[ {\left(Y-\text{E}{\left[Y \mid X\right]}\right)} {\left(\text{E}{\left[Z \mid X\right]}-\text{E}{\left[Z\right]}\right)} \mid X=x \right]} \cdot\text{p}(X=x)\, dx \\&= \int_{x \in \mathcal{R}(X)} {\left(\text{E}{\left[Z \mid X=x\right]}-\text{E}{\left[Z\right]}\right)} \cdot\text{E}{\left[Y-\text{E}{\left[Y \mid X=x\right]} \mid X=x\right]} \cdot\text{p}(X=x)\, dx \\&= 0 \end{aligned} \]

and similarly:

\[ \text{E}{\left[{\left(\text{E}{\left[Y \mid X\right]}-\text{E}{\left[Y\right]}\right)}{\left(Z-\text{E}{\left[Z \mid X\right]}\right)}\right]}=0. \]

Hence:

\[ \begin{aligned} \text{Cov}{\left(Y,Z\right)} &= \text{E}{\left[{\left(Y-\text{E}{\left[Y \mid X\right]}\right)}{\left(Z-\text{E}{\left[Z \mid X\right]}\right)}\right]} + \text{E}{\left[{\left(\text{E}{\left[Y \mid X\right]}-\text{E}{\left[Y\right]}\right)}{\left(\text{E}{\left[Z \mid X\right]}-\text{E}{\left[Z\right]}\right)}\right]} \\&= \text{E}{\left[\text{Cov}{\left(Y,Z \mid X\right)}\right]} + \text{Cov}{\left(\text{E}{\left[Y \mid X\right]}, \text{E}{\left[Z \mid X\right]}\right)} \end{aligned} \]

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Lemma 1 (The covariance of a variable with itself is its variance) For any random variable \(X\):

\[\text{Cov}{\left(X,X\right)} = \text{Var}{\left(X\right)}\]

ProofProof

Proof. \[ \begin{aligned} \text{Cov}{\left(X,X\right)} &= \text{E}{\left[XX\right]} - \text{E}{\left[X\right]}\text{E}{\left[X\right]} \\&= \text{E}{\left[X^2\right]} - {\left(\text{E}{\left[X\right]}\right)}^2 \\ &= \text{Var}{\left(X\right)} \end{aligned} \]

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Definition 22 (Variance/covariance of a \(p \times 1\) random vector) For a \(p \times 1\) dimensional random vector \(\tilde{X}\),

\[ \begin{aligned} \text{Var}{\left(\tilde{X}\right)} &\stackrel{\text{def}}{=}\text{Cov}{\left(\tilde{X}\right)} \\ &\stackrel{\text{def}}{=}\text{E}{\left[{\left(\tilde{X}- \text{E}\tilde{X}\right)} {{\left(\tilde{X}- \text{E}\tilde{X}\right)}}^{\top}\right]} \end{aligned} \]

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Theorem 24 (Elements of the variance-covariance matrix are pairwise covariances) For a \(p \times 1\) random vector \(\tilde{X}= {(X_1, \ldots, X_p)}^{\top}\), the \((i,j)\)-th element of \(\text{Var}{\left(\tilde{X}\right)}\) is \(\text{Cov}{\left(X_i, X_j\right)}\):

\[ \text{Var}{\left(\tilde{X}\right)}= \begin{pmatrix} \text{Var}{\left(X_1\right)} & \text{Cov}{\left(X_1, X_2\right)} & \cdots & \text{Cov}{\left(X_1, X_p\right)} \\ \text{Cov}{\left(X_2, X_1\right)} & \text{Var}{\left(X_2\right)} & \cdots & \text{Cov}{\left(X_2, X_p\right)} \\ \vdots & \vdots & \ddots & \vdots \\ \text{Cov}{\left(X_p, X_1\right)} & \text{Cov}{\left(X_p, X_2\right)} & \cdots & \text{Var}{\left(X_p\right)} \end{pmatrix} \]

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ProofProof

Proof. Let \(\mu_i = \text{E}{\left[X_i\right]}\) for \(i = 1, \ldots, p\), so \(\text{E}\tilde{X}= {(\mu_1, \ldots, \mu_p)}^{\top}\). By Definition 22:

\[ \begin{aligned} \text{Var}{\left(\tilde{X}\right)} &= \text{E}{\left[ {\left(\tilde{X}- \text{E}\tilde{X}\right)} {{\left(\tilde{X}- \text{E}\tilde{X}\right)}}^{\top} \right]} \\ &= \text{E}{\left[ \begin{pmatrix}X_1 - \mu_1 \\ \vdots \\ X_p - \mu_p\end{pmatrix} \begin{pmatrix}X_1 - \mu_1 & \cdots & X_p - \mu_p\end{pmatrix} \right]} \\ &= \text{E}{\left[ \begin{pmatrix} (X_1 - \mu_1)(X_1 - \mu_1) & \cdots & (X_1 - \mu_1)(X_p - \mu_p) \\ \vdots & \ddots & \vdots \\ (X_p - \mu_p)(X_1 - \mu_1) & \cdots & (X_p - \mu_p)(X_p - \mu_p) \end{pmatrix} \right]} \\ &= \begin{pmatrix} \text{E}{\left[(X_1 - \mu_1)(X_1 - \mu_1)\right]} & \cdots & \text{E}{\left[(X_1 - \mu_1)(X_p - \mu_p)\right]} \\ \vdots & \ddots & \vdots \\ \text{E}{\left[(X_p - \mu_p)(X_1 - \mu_1)\right]} & \cdots & \text{E}{\left[(X_p - \mu_p)(X_p - \mu_p)\right]} \end{pmatrix} \\ &= \begin{pmatrix} \text{Cov}{\left(X_1, X_1\right)} & \cdots & \text{Cov}{\left(X_1, X_p\right)} \\ \vdots & \ddots & \vdots \\ \text{Cov}{\left(X_p, X_1\right)} & \cdots & \text{Cov}{\left(X_p, X_p\right)} \end{pmatrix} \\ &= \begin{pmatrix} \text{Var}{\left(X_1\right)} & \cdots & \text{Cov}{\left(X_1, X_p\right)} \\ \vdots & \ddots & \vdots \\ \text{Cov}{\left(X_p, X_1\right)} & \cdots & \text{Var}{\left(X_p\right)} \end{pmatrix} \end{aligned} \]

where:

• the step from the third to fourth line uses Definition 15,
• the step from the fourth to fifth line uses Definition 21, and
• the last step uses Lemma 1.

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Theorem 25 (Alternate expression for variance of a random vector) \[ \begin{aligned} \text{Var}{\left(\tilde{X}\right)} &= \text{E}{\left[\tilde{X}{\tilde{X}}^{\top}\right]} - {\left(\text{E}\tilde{X}\right)} {{\left(\text{E}\tilde{X}\right)}}^{\top} \end{aligned} \]

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ProofProof

Proof. \[ \begin{aligned} \text{Var}{\left(\tilde{X}\right)} &= \text{E}{\left[ {\left(\tilde{X}- \text{E}\tilde{X}\right)} {{\left(\tilde{X}- \text{E}\tilde{X}\right)}}^{\top} \right]} \\ &= \text{E}{\left[ \tilde{X}{\tilde{X}}^{\top} - \tilde{X}{{\left(\text{E}\tilde{X}\right)}}^{\top} - {\left(\text{E}\tilde{X}\right)} {\tilde{X}}^{\top} + {\left(\text{E}\tilde{X}\right)} {{\left(\text{E}\tilde{X}\right)}}^{\top} \right]} \\ &= \text{E}{\left[\tilde{X}{\tilde{X}}^{\top}\right]} - {\left(\text{E}\tilde{X}\right)} {{\left(\text{E}\tilde{X}\right)}}^{\top} - {\left(\text{E}\tilde{X}\right)} {{\left(\text{E}\tilde{X}\right)}}^{\top} + {\left(\text{E}\tilde{X}\right)} {{\left(\text{E}\tilde{X}\right)}}^{\top} \\ &= \text{E}{\left[\tilde{X}{\tilde{X}}^{\top}\right]} - {\left(\text{E}\tilde{X}\right)} {{\left(\text{E}\tilde{X}\right)}}^{\top} \end{aligned} \]

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Theorem 26 (Variance of a linear combination) For any vector of random variables \(\tilde{X}= (X_1, \ldots, X_n)\) and corresponding vector of constants \(\tilde{a}= (a_1, ... ,a_n)\), the variance of their linear combination is:

\[ \begin{aligned} \text{Var}{\left(\tilde{a}\cdot \tilde{X}\right)} &= \text{Var}{\left(\sum_{i=1}^na_i X_i\right)} \\ &= \tilde{a}^{\top} \text{Var}{\left(\tilde{X}\right)} \tilde{a} \\ &= \sum_{i=1}^n\sum_{j=1}^n a_i a_j \text{Cov}{\left(X_i,X_j\right)} \end{aligned} \]

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ProofProof

Proof. Left to the reader…

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Corollary 3 For any two random variables \(X\) and \(Y\) and scalars \(a\) and \(b\):

\[\text{Var}{\left(aX + bY\right)} = a^2 \text{Var}{\left(X\right)} + b^2 \text{Var}{\left(Y\right)} + 2(a \cdot b) \text{Cov}{\left(X,Y\right)}\]

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ProofProof

Proof. Apply Theorem 26 with \(n=2\), \(X_1 = X\), and \(X_2 = Y\).

Or, see https://statproofbook.github.io/P/var-lincomb.html

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Definition 23 (homoskedastic, heteroskedastic) A random variable \(Y\) is homoskedastic (with respect to covariates \(X\)) if the variance of \(Y\) does not vary with \(X\):

\[\text{Var}(Y|X=x) = \sigma^2, \forall x\]

Otherwise it is heteroskedastic.

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Definition 24 (Statistical independence) A set of random variables \(X_1, \ldots, X_n\) are statistically independent if their joint probability is equal to the product of their marginal probabilities:

\[\Pr(X_1=x_1, \ldots, X_n = x_n) = \prod_{i=1}^n{\Pr(X_i=x_i)}\]

Tip

The symbol for independence, \(\perp\!\!\!\perp\), is essentially just \(\prod\) upside-down. So the symbol can remind you of its definition (Definition 24).

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Definition 25 (Conditional independence) A set of random variables \(Y_1, \ldots, Y_n\) are conditionally statistically independent given a set of covariates \(X_1, \ldots, X_n\) if the joint probability of the \(Y_i\)s given the \(X_i\)s is equal to the product of their marginal probabilities:

\[\Pr(Y_1=y_1, \ldots, Y_n = y_n|X_1=x_1, \ldots, X_n = x_n) = \prod_{i=1}^n{\Pr(Y_i=y_i|X_i=x_i)}\]

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Definition 26 (Identically distributed) A set of random variables \(X_1, \ldots, X_n\) are identically distributed if they have the same range \(\mathcal{R}(X)\) and if their marginal distributions \(\text{P}(X_1=x_1), ..., \text{P}(X_n=x_n)\) are all equal to some shared distribution \(\text{P}(X=x)\):

\[ \forall i\in {\left\{1:n\right\}}, \forall x \in \mathcal{R}(X): \text{P}(X_i=x) = \text{P}(X=x) \]

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Definition 27 (Conditionally identically distributed) A set of random variables \(Y_1, \ldots, Y_n\) are conditionally identically distributed given a set of covariates \(X_1, \ldots, X_n\) if \(Y_1, \ldots, Y_n\) have the same range \(\mathcal{R}(X)\) and if the distributions \(\text{P}(Y_i=y_i|X_i =x_i)\) are all equal to the same distribution \(\text{P}(Y=y|X=x)\):

\[ \text{P}(Y_i=y|X_i=x) = \text{P}(Y=y|X=x) \]

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Definition 28 (Independent and identically distributed) A set of random variables \(X_1, \ldots, X_n\) are independent and identically distributed (shorthand: “\(X_i\ \text{iid}\)”) if they are statistically independent and identically distributed.

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Definition 29 (Conditionally independent and identically distributed) A set of random variables \(Y_1, \ldots, Y_n\) are conditionally independent and identically distributed (shorthand: “\(Y_i | X_i\ \text{ciid}\)” or just “\(Y_i |X_i\ \text{iid}\)”) given a set of covariates \(X_1, \ldots, X_n\) if \(Y_1, \ldots, Y_n\) are conditionally independent given \(X_1, \ldots, X_n\) and \(Y_1, \ldots, Y_n\) are identically distributed given \(X_1, \ldots, X_n\).

3.6 The Central Limit Theorem

The sum of many independent or nearly-independent random variables with small variances (relative to the number of RVs being summed) produces bell-shaped distributions.

For example, consider the sum of five dice (Figure 4).

library(dplyr)
dist = 
  expand.grid(1:6, 1:6, 1:6, 1:6, 1:6) |> 
  rowwise() |>
  mutate(total = sum(c_across(everything()))) |> 
  ungroup() |> 
  count(total) |> 
  mutate(`p(X=x)` = n/sum(n))

library(ggplot2)

dist |> 
  ggplot() +
  aes(x = total, y = `p(X=x)`) +
  geom_col() +
  xlab("sum of dice (x)") +
  ylab("Probability of outcome, Pr(X=x)") +
  expand_limits(y = 0)

  
  

Figure 4: Distribution of the sum of five dice

In comparison, the outcome of just one die is not bell-shaped (Figure 5).

library(dplyr)
dist = 
  expand.grid(1:6) |> 
  rowwise() |>
  mutate(total = sum(c_across(everything()))) |> 
  ungroup() |> 
  count(total) |> 
  mutate(`p(X=x)` = n/sum(n))

library(ggplot2)

dist |> 
  ggplot() +
  aes(x = total, y = `p(X=x)`) +
  geom_col() +
  xlab("sum of dice (x)") +
  ylab("Probability of outcome, Pr(X=x)") +
  expand_limits(y = 0)

  
  

Figure 5: Distribution of the outcome of one die

What distribution does a single die have?

Answer: discrete uniform on 1:6.

4 Additional resources

• Miller (2017)

References

Dobson, Annette J, and Adrian G Barnett. 2018. An Introduction to Generalized Linear Models. 4th ed. CRC press. https://doi.org/10.1201/9781315182780.

Kalbfleisch, John D, and Ross L Prentice. 2011. The Statistical Analysis of Failure Time Data. John Wiley & Sons.

Klein, John P, and Melvin L Moeschberger. 2003. Survival Analysis: Techniques for Censored and Truncated Data. Vol. 1230. Springer. https://link.springer.com/book/10.1007/b97377.

Kleinbaum, David G, and Mitchel Klein. 2012. Survival Analysis: A Self-Learning Text. 3rd ed. Springer. https://link.springer.com/book/10.1007/978-1-4419-6646-9.

Miller, Steven J. 2017. The Probability Lifesaver : All the Tools You Need to Understand Chance. A Princeton Lifesaver Study Guide. Princeton University Press. https://press.princeton.edu/books/hardcover/9780691149547/the-probability-lifesaver.

Rothman, Kenneth J., Timothy L. Lash, Tyler J. VanderWeele, and Sebastien Haneuse. 2021. Modern Epidemiology. Fourth edition. Wolters Kluwer.

Vittinghoff, Eric, David V Glidden, Stephen C Shiboski, and Charles E McCulloch. 2012. Regression Methods in Biostatistics: Linear, Logistic, Survival, and Repeated Measures Models. 2nd ed. Springer. https://doi.org/10.1007/978-1-4614-1353-0.

Footnotes

1. for example in Dobson and Barnett (2018), Vittinghoff et al. (2012), Klein and Moeschberger (2003), and Kleinbaum and Klein (2012)

2. for example, in Rothman et al. (2021) and Kalbfleisch and Prentice (2011)