1. 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

2. 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

3. 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

4. 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

5. 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)),

6. Geometric Distribution Mean: Variance:

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

8. 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

9. Negative Binomial Distribution If W is the number of independent Bernoulli trials (Ber(p)) required to get the r th success, then W has a negative binomial distribution,

10. Geometric Distribution Mean: Variance:

11. 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

12. Example Suppose that typographical errors occur at a rate of ½ per page Find the probability of getting 3 mistakes in a given page

13. 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:

14. Example Suppose that typographical errors occur at a rate of ½ per page Find the probability of getting 3 mistakes in a given page

15. 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?

16. 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

17. Chapter 5 Continuous Random Variables Not all outcomes can be listed (e.g., {w 1, w 2, …,}) 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

18. Continuous Random Variables Axioms of probability must still hold Events are usually expressed in intervals for a continuous random variable

19. 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:

20. 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

21. Properties Probability of an interval:

22. 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)

23. Example Suppose X has cdf, Find P(X<1/2) Find P(.5<X<3)

24. 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

25. Distribution Functions and Densities Therefore, That is, the probability of an interval is the area under the density curve

26. 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)

27. Properties of the Density