Other Distributions

Class: STAT 131 Subject: probability Date: 2025-01-29 Teacher: Prof. Marcela

Hypergeometric Distribution

Formula

$P(X=K)=\frac{\left(\genfrac{}{}{0ex}{}{s}{k}\right)\left(\genfrac{}{}{0ex}{}{f}{n-k}\right)}{\left(\genfrac{}{}{0ex}{}{s+f}{n}\right)}$

• Population = $s+f$
• Sample = $n$ ($k$ and $n-k$)

Example

• Items in a population are classified using two sets of tags: success and failure, or tagged not tagged, or two ball colors in an urn.
• Example: You have an urn with $N=10$ balls, $s=6$ that are tagged as winners and $f=4$ that are not. You take a sample of $n=3$ balls, without replacement. What is the probability of observing $k=2$ tagged balls on your sample?

Explanation

$P(X=2)=\frac{\left(\genfrac{}{}{0ex}{}{6}{2}\right)\left(\genfrac{}{}{0ex}{}{4}{3-2}\right)}{\left(\genfrac{}{}{0ex}{}{6+4}{3}\right)}=0.5$ $P(X\le 2)=P(X=0)+P(X=1)+P(X=2)$ $=$ $\frac{\left(\genfrac{}{}{0ex}{}{6}{0}\right)\left(\genfrac{}{}{0ex}{}{4}{3-0}\right)}{\left(\genfrac{}{}{0ex}{}{6+4}{3}\right)}+\frac{\left(\genfrac{}{}{0ex}{}{6}{1}\right)\left(\genfrac{}{}{0ex}{}{4}{3-1}\right)}{\left(\genfrac{}{}{0ex}{}{6+4}{3}\right)}+\frac{\left(\genfrac{}{}{0ex}{}{6}{2}\right)\left(\genfrac{}{}{0ex}{}{4}{3-2}\right)}{\left(\genfrac{}{}{0ex}{}{6+4}{3}\right)}=0.03333+0.03+0.05=0.83333$

Uniform Distribution

Formula

$P(X=x)=\frac{1}{\mid C\mid }$

Example

• A raffle. You have 100 possible numbers, $X$ is a random variable that represents
  the number that you select for the raffle, randomly $P(X=8)=\frac{1}{100}$

PMF and CDF

Example

• Given the following: a) What is the support of $x$? { 0, 1, 2} b) Find $P(x\ge 1.5)$ $P(x\ge 1.5)=P(X=2)=\frac{1}{2}$ c) Find $P(0<x<2)$ $P(0<x<2)=P(x=1)=\frac{1}{3}$ d) Find $P(x=0\mid x<2)$ $\frac{P(x=0\cap x<2)}{P(x<2)}=\frac{P(x=0)}{P(x<2)}=\frac{\frac{1}{2}}{\frac{5}{6}}=3/5$