1 / 27

Today

Understand hypergeometric, geometric, negative binomial, and Poisson distributions along with continuous random variables. Learn properties, examples, and applications.

dyre
Download Presentation

Today

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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 rth 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 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

  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

More Related