## Introduction to Probability Models

Lecture 14

Qi Wang, Department of Statistics

Sep 21, 2018

## Reminders

- The second homework is due
**NOW**

## Named Random Variables

- Bernoulli
- Binomial
- Hypergeometric
- Poisson
- Geometric
- Negative Binomial

## Bernoulli Distribution

- The probability distribution of a random variable which takes the value 1 with probability $p$ and the value 0 with probability $q=1-p$
- The outcome is
**YES or NO**, **SUCCESS or FAILURE**, **1 or 0** $\cdots$

## Bernoulli Distribution

- $X \sim Bern(p)$
**Support: **$\{0, 1\}$
**Parameter: ** p
**PMF: **$P_X(x) = p^x (1-p)^{1-x}$
**Expected Value: ** $p$
**Variance: **$p(1-p)$

## Example 1

In Eric’s STAT 225 class, 80% of the students passed on Exam 1. If we were to pick a student at random and asked them whether or not they passed, let X be the number of students who passed.
What type of random variable is this? How do you know? Additionally,
write down the pmf, the expected value, and the variance of X.

## Binomial Distribution

- $X\sim Binomial(n, p)$
- The total number of successes in a sequence of n independent Bernoulli experiments, with a success rate p
**Support: **$\{0, 1, 2, \cdots, n\}$
**Expected Value: **$np$
**Variance: **$np(1-p)$

## Example 2

Now pick 10 students from Eric’s class, with the same probability of having passed. Let X be the total number of students who passed.
What type of random variable is this? What values can X take? Please write down the pmf, the expected value, and the variance of X.

### Relationship between Bernoulli Distribution and Binomial Distribution

**Theorem: **Let $X_1, X_2, \cdots, X_n$ be independent Bernoulli random variables, each with the same parameter p.
Then the sum $X= X_1 + X_2 + \cdots + X_n$ is a binomial random variable with parameters $n$ and $p$