## Introduction to Probability Models

Lecture 14

Qi Wang, Department of Statistics

Sep 21, 2018

## Reminders

1. 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 $p$