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)),
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,
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
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•
•
• 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
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)
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 )()(