The geometric distribution

The original notes introduce the geometric family from the geometric series

\sum_{k=0}^{\infty}\frac{1}{2^{k+1}}=1.

Thus p_K(k)\equiv2^{-(k+1)}, for k=0,1,2,\ldots, is a probability mass function (PMF) with countably infinite support.

Code

k <- 0:9
mass <- 2^(-(k + 1))
plot(k, mass, type = "h", lwd = 5, lend = 1, col = "#B2182B",
     xlab = "k", ylab = "Probability",
     main = "PMF defined by a geometric series", bty = "l")
points(k, mass, pch = 19, col = "#B2182B")

Definition and convention

Let K count failures before the first success in independent Bernoulli trials with common success probability \pi\in(0,1]. We define

K\sim\operatorname{Geometric}(\pi)

p_K(k) \equiv\mathbb P(K=k) =(1-\pi)^k\pi, \qquad k=0,1,2,\ldots.

This is the convention used by dgeom in R. If T counts total trials through the first success, then T\equiv K+1 and

\mathbb P(T=t)=(1-\pi)^{t-1}\pi, \qquad t=1,2,\ldots.

Constructing the PMF from trial sequences

Suppose we sample word tokens until the first loanword, with constant loanword probability \pi=.20. The event K=k requires k nonloanwords followed by one loanword. Independence gives

\mathbb P(K=k)=(.80)^k(.20).

Thus

\begin{aligned} p_K(0)&=.20,\\ p_K(1)&=.80(.20)=.16,\\ p_K(2)&=.80^2(.20)=.128,\\ p_K(3)&=.80^3(.20)=.1024. \end{aligned}

The event K\geq4 requires four initial failures, so

\mathbb P(K\geq4)=.80^4=.4096

and \mathbb P(T\leq4)=1-.4096=.5904.

The original parameter sequence

The following graphs reproduce the \pi=.5, .1, and .9 sequence from the original notes.

Code

plot_geometric <- function(probability) {
  k <- 0:9
  mass <- dgeom(k, prob = probability)
  plot(k, mass, type = "h", lwd = 5, lend = 1, col = "#B2182B",
       xlab = "Failures before the first success", ylab = "Probability",
       main = sprintf("PMF of Geometric(%.1f)", probability), bty = "l")
  points(k, mass, pch = 19, col = "#B2182B")
}

plot_geometric(.5)

Code

plot_geometric(.1)

Code

plot_geometric(.9)

For every k\geq0,

\frac{p_K(k+1)}{p_K(k)} =1-\pi<1.

Thus every nondegenerate geometric PMF is strictly decreasing. Its mode is K=0. If L\equiv K+1 represents positive word length, its mode is necessarily L=1. The negative-binomial page compares this restriction with the empirical distribution of phoneme counts in CMUdict.

Under this convention,

\mathbb E[K]=\frac{1-\pi}{\pi}, \qquad \operatorname{Var}(K)=\frac{1-\pi}{\pi^2}.

--- title: "The geometric distribution" execute: enabled: true echo: true warning: false message: false --- The original notes introduce the geometric family from the geometric series $$ \sum_{k=0}^{\infty}\frac{1}{2^{k+1}}=1. $$ Thus $p_K(k)\equiv2^{-(k+1)}$, for $k=0,1,2,\ldots$, is a probability mass function (PMF) with countably infinite support. ```{r} k <- 0:9 mass <- 2^(-(k + 1)) plot(k, mass, type = "h", lwd = 5, lend = 1, col = "#B2182B", xlab = "k", ylab = "Probability", main = "PMF defined by a geometric series", bty = "l") points(k, mass, pch = 19, col = "#B2182B") ``` ## Definition and convention Let $K$ count failures before the first success in independent Bernoulli trials with common success probability $\pi\in(0,1]$. We define $$ K\sim\operatorname{Geometric}(\pi) $$ by $$ p_K(k) \equiv\mathbb P(K=k) =(1-\pi)^k\pi, \qquad k=0,1,2,\ldots. $$ This is the convention used by `dgeom` in R. If $T$ counts total trials through the first success, then $T\equiv K+1$ and $$ \mathbb P(T=t)=(1-\pi)^{t-1}\pi, \qquad t=1,2,\ldots. $$ ## Constructing the PMF from trial sequences Suppose we sample word tokens until the first loanword, with constant loanword probability $\pi=.20$. The event $K=k$ requires $k$ nonloanwords followed by one loanword. Independence gives $$ \mathbb P(K=k)=(.80)^k(.20). $$ Thus $$ \begin{aligned} p_K(0)&=.20,\\ p_K(1)&=.80(.20)=.16,\\ p_K(2)&=.80^2(.20)=.128,\\ p_K(3)&=.80^3(.20)=.1024. \end{aligned} $$ The event $K\geq4$ requires four initial failures, so $$ \mathbb P(K\geq4)=.80^4=.4096 $$ and $\mathbb P(T\leq4)=1-.4096=.5904$. ## The original parameter sequence The following graphs reproduce the $\pi=.5$, $.1$, and $.9$ sequence from the original notes. ```{r} plot_geometric <- function(probability) { k <- 0:9 mass <- dgeom(k, prob = probability) plot(k, mass, type = "h", lwd = 5, lend = 1, col = "#B2182B", xlab = "Failures before the first success", ylab = "Probability", main = sprintf("PMF of Geometric(%.1f)", probability), bty = "l") points(k, mass, pch = 19, col = "#B2182B") } plot_geometric(.5) plot_geometric(.1) plot_geometric(.9) ``` For every $k\geq0$, $$ \frac{p_K(k+1)}{p_K(k)} =1-\pi<1. $$ Thus every nondegenerate geometric PMF is strictly decreasing. Its mode is $K=0$. If $L\equiv K+1$ represents positive word length, its mode is necessarily $L=1$. The [negative-binomial page](negative-binomial-distribution.qmd) compares this restriction with the empirical distribution of phoneme counts in CMUdict. Under this convention, $$ \mathbb E[K]=\frac{1-\pi}{\pi}, \qquad \operatorname{Var}(K)=\frac{1-\pi}{\pi^2}. $$