Suppose that you have three coins. Coin A is fair, coin B shows heads with probability 0.6 and coin C shows heads with probability 0.8.

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1 Suppose that you have three coins. Coin A is fair, coin B shows heads with probability 0.6 and coin C shows heads with probability 0.8. Coin A is flipped until a head appears, then coin B is flipped until a head appears and finally coin C is flipped until a head appears. What is the expected total number of tosses to get the three heads? Let X A = # flips to get heads on coin A Let X B = # flips to get heads on coin B Let X C = # flips to get heads on coin C E( X ) A = 2 E( X ) B = = 10 6 E( X ) C = = 10 8 E X ( ) = E X A ( ) + E X B ( ) + E X C ( ) = 59 12

2 The Hypergeometric Random Variable Suppose there is a box with N items in it and g of them are good and the rest are bad. We choose a random sample of n of the items and let X denote the number of the sample that are good. P(X = k) = g N g k n k N n, 0 k g

3 The Hypergeometric Random Variable g N g k n k P(X = k) =, 0 k g N n What is the mean and the variance for a hypergeometric random variable? µ = E( X) = ng N, σ 2 = Var(X) = n g N N g N N n N 1

4 What is the value of g b g k n k = 1 N k=0 k What is the mean and the variance for a hypergeometric random variable. µ = E( X) = ng N, σ 2 = Var(X) = n g N N g N N n N 1

5 p(k) = P( X = k) = µ = E( X) = ng g N g k n k, 0 k g N k N, σ 2 = Var(X) = n g N N g N N n N 1 p = g N, q = N g N µ = np, σ 2 = npq N n N 1 Note lim n N n N 1 = 1

6 A batch of 50 fuses has 10 bad and 40 good. A sample of 6 is chosen to test. What is the probability exactly 3 are bad? P( X = 3) = X is approximately b(n, p) for p = 0.2, n = 6 ( ) = 6 3 P X = ( ) 3 (0.8) 3 = =

7 In the South Carolina Powerball Lottery, each ticket contains five numbers chosen from the numbers You win if the lottery system chooses the exact same five numbers. Suppose that you play one ticket in the South Carolina Lottery. Note that X = number of winning numbers on your ticket is a hypergeometric random variable. There are 55 items total and 5 are good. N = 55, g = 5. We choose a sample of size 5. n = 5. P( X = k) = 5 50 k 5 k 55 5, 0 k 5

8 In the South Carolina Powerball Lottery, each ticket contains five numbers chosen from the numbers You win if the lottery system chooses the exact same five numbers. Suppose that you play one ticket in the South Carolina Lottery. (a). What is the probability that none of the numbers you select will appear in the winning number? P( X = 0) = =

9 In the South Carolina Powerball Lottery, each ticket contains five numbers chosen from the numbers You win if the lottery system chooses the exact same five numbers. Suppose that you play one ticket in the South Carolina Lottery. (b). What is the probability that exactly four of the numbers you select will appear in the winning number? P( X = 4) = = 5 50 =

10 In the South Carolina Powerball Lottery, each ticket contains five numbers chosen from the numbers You win if the lottery system chooses the exact same five numbers. Suppose that you play one ticket in the South Carolina Lottery. (b). What is the probability that your ticket contains at most one winning number? P( X = 0) + P( X = 1) =

11 Suppose that X is a geometric random variable with probability p of success and q of failure on a given trial. Determine, as a simple function of q, the probability that X takes on an even value. ( ) = P( X = 2k) P X is even = pq 2k 1 = p k=1 k=1 = p( q + q 3 + q 5 + q 7 + ) = p k=1 q q 2k 1 1 q 2 = p q 1 q 2 = p q 1 q ( )( 1+ q) = q 1+ q p

12 k=0 x k = e x

13 So now suppose that X is a discrete random variable with the distribution function: p(k) = P( X = k) = e λ λ k, k = 0, 1, 2 X is said to be a Poisson random variable. ( ) = kp(k) = E X e λ λ k k k=0 λ k = e λ ( k 1)! k=0 k=0 λ k = λ e λ ( k 1)! k=0 k 1 k=1 = λe λ λ k 1 ( k 1)! = λe λ e λ = λ

14 What kinds of phenomena are often modeled by a Poisson random variable? Number of accidents at an intersection. Number of errors on a page of a book. Number of flaws in a foot of aluminum foil. Number of plane crashes in a given year. In general, number of events over a period of time/distance.

15 Suppose that customers enter a store at a rate of about 5 per hour. What is the probability that exactly 3 customers enter the store in the next half hour? This is modeled by a Poisson random variable with mean 5. λ = 5 Let X denote the number of customers that enter the store during a given hour. P( X = k) = e 5 5 k P( X = 3) = e ! = 125 6e

16 n k = n n 1 n n n ( ) n 2 ( ) n k +1 ( ) nk There are k terms here. If k is much smaller than n, then n - k ~ n If n is much larger than k, n k nk

17 So now suppose that X is a b(n,p) binomial random variable. Let λ = np p = λ n ( ) = n k P X = k pk q n k nk = λ k 1 λ n n k= λ k λ n k 1 λ n 1 λ n 1 λ n n k n k e λ 1 λ k e λ 1 = e λ λ k

18 If X is a random variable then the k th So, the first moment is E(X), the mean. moment of X is E( X k ) The second moment is E(X 2 ), used for finding the variance. The moment generating function for X is E(e tx ) < ( ) M X (t) = E e tx provided for t in an (-h, h) for some h > 0. M ( X 0) = 1 M X ( 0) = E( X) M X ( 0) = E( X 2 ) M X (k ) M X ( 0) = E( X k ) t ( ) = E e tx ( ) = 1+ tm 1 + t 2 2 M 2 + t 3 6 M 3 + t k M k +