Lecture 10

Zhihua (Sophia) Su

University of Florida

Jan 30, 2015

STA 4321/5325 Introduction to Probability 1

Agenda

Binomial Random VariableGeometric Random Variable

Reading assignment: Chapter 3: 3.4, 3.5

STA 4321/5325 Introduction to Probability 2

Binomial Random Variable

Recall that a Bernoulli trial or a Bernoulli experiment has only2 outcomes. The associated Bernoulli random variable X takesthe value 0 if the outcome is a success, and 1 if the outcome is afailure. Let P (X = 1) = p and P (X = 0) = 1− p, i.e., p is theprobability of success for the experiment.

E(X) = p and V (X) = p(1− p).

STA 4321/5325 Introduction to Probability 3

Binomial Random Variable

Suppose an experiment consist of n independent Bernoulli trials.For example, toss a coin 1000 times (n = 1000) or inspect 1000items for being defective (n = 1000).

Let X = # of success in the n trials.

Yi =

{1 if ith trial is success,0 if ith trial is failure.

X =∑n

i=1 Yi.

STA 4321/5325 Introduction to Probability 4

Binomial Random Variable

DefinitionA random variable X is said to be a Binomial random variablewith parameter n and p if(i) X is the number of successes in n independent Bernoulli

trials.(ii) The probability of success in each trial is p.

STA 4321/5325 Introduction to Probability 5

Binomial Random Variable

Let us calculate the probability mass function of a Binomialrandom variable.

The probability mass function of a Binomial random variable is

pX(x) =

(nx

)px(1− p)n−x, for x = 0, 1, · · · , n.

STA 4321/5325 Introduction to Probability 6

Binomial Random Variable

Prove that the expected value of a Binomial random variable Xis E(X) = np.

STA 4321/5325 Introduction to Probability 7

Binomial Random Variable

Fact: If Y1, Y2, . . ., Yn are random variables arising out ofindependent experiments, then

V

(n∑

i=1

Yi

)=

n∑i=1

V (Yi).

Hence V (X) = np(1− p).

STA 4321/5325 Introduction to Probability 8

Binomial Random Variable

A hospital has to install n generators with the requirement thatwith probability 0.99 at least one generator should be working atany given time. Suppose that the probability that an generatoris operating correctly is 0.95. What is the minimum value of n?

STA 4321/5325 Introduction to Probability 9

Geometric Random Variable

Consider an experiment which consists of repeating independentBernoulli trials until a success is obtained. Assume that theprobability of success in each independent trial is p.

Let X = # of the trial on which the first success occurs.

X = Range(X) = {1, 2, . . .}

STA 4321/5325 Introduction to Probability 10

Geometric Random Variable

Find the probability mass function and distributive function ofa geometric random variable X.

STA 4321/5325 Introduction to Probability 11