Today

• Today: Finish Chapter 4, Start Chapter 5

• Reading: – Chapter 5 (not 5.12)

– Important Sections From Chapter 4• 4-1-4.4 (excluding the negative hypergeometric distribution)

• 4.6

– Suggested problems: 5.1, 5.2, 5.3, 5.15, 5.25, 5.33, 5.38, 5.47, 5.53, 5.62

Hypergeometric Distribution

• When M/N is essentially constant, the hypergeometric probabilities can be approximated by using the binomial distribution

• Example– Suppose 40% of voters of the 500,000 voters in a city are Democrats

– A poll of 500 voters is done

– What is the probability that 50% of voters claim to be Democrats

Example

• In the game Monopoly, where players roll two dice, a player can end up in “jail”

• To get out of jail, the player must roll two of a kind to get out of jail

• Find the probability that a player rolls a “doubles” on their turn

Example

• If Z is the random variable denoting the number of turns required to get out of jail, what is the probability function for Z

Geometric Distribution

• If Z is the number of independent Bernoulli trials (Ber(p)) required to get a success, then Z has a geometric distribution (Z~Geo(p)),

Geometric Distribution

• Mean:

• Variance:

Example

• In Monopoly, what is the expected number of turns required to get out of jail?

Example

• Suppose an archer hits a bull’s-eye once in every 10 tries on average

• Find the probability she hits her first bull’s-eye on the 11 trial

• Find the probability she hits her third bull’s-eye on the 15 trial

Negative Binomial Distribution

• If W is the number of independent Bernoulli trials (Ber(p)) required to get the rth success, then W has a negative binomial distribution,

Geometric Distribution

• Mean:

• Variance:

Example

• Suppose an archer hits a bull’s-eye once in every 10 tries on average

• Find the probability she hits her third bull’s-eye on the 15 trial

• Find the expected number of trials required to get the third bull’s-eye

Example

• Suppose that typographical errors occur at a rate of ½ per page

• Find the probability of getting 3 mistakes in a given page

Poisson Distribution

• If X is a random variable denoting the number (the count) of events in any region of fixed size, and λ is the rate at which these events occur, then the probability function for X is:

Example

• Suppose that typographical errors occur at a rate of ½ per page

• Find the probability of getting 3 mistakes in a given page

Example

• Find the expected number of errors on a given page

• What is the probability distribution of the number of errors in a 20 page paper?

Example

• A study on the number of calls to a wrong number at a payphone in a large train terminal was conducted (Thornedike, 1926)

• According to the study, the number of calls to wrong numbers in a one minute interval follows a Poisson distribution with parameter λ=1.20

• Find the probability that the number of wrong numbers in a 1 minute interval is two

• Find the probability that the number of wrong numbers in a 1 minute interval is between two and 4

Chapter 5Continuous Random Variables

• Not all outcomes can be listed (e.g., {w1, w2, …,}) as in the case of discrete random variable

• Some random variables are continuous and take on infinitely many values in an interval

• E.g., height of an individual

Continuous Random Variables

• Axioms of probability must still hold

•

•

•

• Events are usually expressed in intervals for a continuous random variable

EEP event any for ;1)(0 1)( P

exclusivemutually are F and E whenever )()()()( FPEPFPEP

Example (Continuous Uniform Distribution)

• Suppose X can take on any value between –1 and 1

• Further suppose all intervals in [-1,1] of length a have the same probability of occurring, then X has a uniform distribution on (-1,1)

• Picture:

Distribution Function of a Continuous Random Variable

• The distribution function of a continuous random variable X is defined as,

• Also called the cumulative distribution function or cdf

Properties

• Probability of an interval:

Example

• Suppose X~U(-1,1), with cdf F(x)=1/2(x+1) for –1<x<1

• Find P(X<0)

• Find P(-.5<X<.5)

• Find P(X=0)

Example

• Suppose X has cdf,

• Find P(X<1/2)

• Find P(.5<X<3)

21 if ,3/)1(

10 if ,3/)(

xx

xxxF

Distribution Functions and Densities

• Suppose that F(x) is the distribution function of a continuous random variable

• If F(x) is differentiable, then its derivative is:

• f(x) is called the density function of X

)()(')( xFdx

dxFxf

Distribution Functions and Densities

• Therefore,

• That is, the probability of an interval is the area under the density curve

a

dxxfaF )()(

Example

• Suppose X~U(0,1), with cdf F(x)=x for –1<x<1

• What is the desnity of X?

• Find P(X<.33)

Properties of the Density