Bernoulli Random Variable

Bernoulli rv, sometimes called a binary rv, is any random variable with only two possible outcomes: $0$ and $1$.

The probability mass function (pmf) is given by:

\[\begin{align} P(X=1)&=p\\ P(X=0)&=1-p\\ \end{align}\]

CDF

\[F(x) = P(X\le x)=\begin{cases} \begin{align} &0&&\text{if }x\lt 0\\ &1-p&&\text{if }0\le x\lt 1\\ &1&&\text{if }1\le x\\ \end{align} \end{cases}\]

Notation: We write $\color{red}{X\sim Bern(p)}$ to indicate that $X$ is a Bernoulli rv with success probability $p$.

Geometric Random Variable

Motivating Example A patient needs a kidney transplant and is waiting for a matching donor.The probability that a random selected donor is a suitable match is $p$.

What is the sample space? What is an appropriate rv? What is the pmf?

\[S=\{1,01,001,0001,\ldots\}\]

Let $X$ = # of donors tested until a match is found.

\[X\in\{1,2,3,\ldots\}\] \[\begin{align} P(X=1)&=p\\ P(X=2)&=(1-p)p\\ P(X=3)&=(1-p)^2p\\ \end{align}\]

PMF

\[\color{red}{P(X=k)=(1-p)^{k-1}p}\quad(k=1,2,3,\ldots)\]

Geometric series

\[a+ar+ar^2+\ldots=\sum_{k=1}^{\infty}ar^{k-1}=\begin{cases} \begin{align} &\frac{a}{1-r}&&\text{if }\vert r\vert\lt1\\ &\text{diverges}&&\text{if }\vert r\vert\ge1\\ \end{align} \end{cases}\]

We have pmf for a geometric r.v.

\[P(X=k)=\underbrace{(1-p)^{k-1}}_{r}\underbrace{p}_{a}\]

Verify that the sum equal $1$:

\[\begin{align} \sum_{k=1}^{\infty}P(X=k)&=\sum_{k=1}^{\infty}(1-p)^{k-1}p\\ &=\frac{p}{1-(1-p)}=1\\ &\text{(note: }r=1-p\lt1) \end{align}\]

A geometric rv consists of independent Bernoulli trials, each with the same probability of success $p$, repeated until the first success is obtained.

  • Each trial is identical, and can result in a success or failure.
  • The probability of success, $p$, is constant from one trial to the next.
  • The trials are independent, so the outcome on any particular trial does not influence the outcome of any other trial.
  • Trials are repeated until the first success.

Summary

♦ Sample space for a geometric rv:

\[S=\{1,01,001,\ldots\}\]

♦ Probability mass function for a geometric rv with probabililty of success $p$:

\[\color{red}{P(X=k)=(1-p)^{k-1}p,\quad k=1,2,3\ldots}\]

♦ Notation: We write $\color{red}{X\sim \text{Geom}(p)}$ to indicate that $X$ is a geometric rv with success probability $p$.