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Today

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 .

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Today

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  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

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