Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
5.5 Geometric Distribution and Negative Binomial Distribution Example: Roll a die repeatedly until number 6 is observed. Let X be the number of times the die has to be rolled to observe a first 6. The possible values for X are 1, 2, 3,……What is P(X = 5)? Solution: {X=5}={First 4 rolls are not 6 and 5th one is a 6.} The probability that a roll does not result in a 6 is 5/6. The probability that a roll results in a 6 is 1/6. P(X = 5)=(5/6)4(1/6) P(X = k)= (5/6)k-1(1/6) , k = 1, 2, … 1 Geometric Distribution If repeated independent trials (Bernoulli trial) can result in a success with probability p and a failure with probability q = 1 – p, then the probability distribution of random variable X, the number of the trial on which the first success occurs is g(x; p) =P(X = x) = qx-1p, x = 1, 2,3, … Theorem 5.4 The mean and variance of a random variable following the geometric distribution g(x; p) are = 1/p , 1 p 2 p 2 2 Example It is known that the probability of being able to log on to a computer from a remote terminal at a given time is 0.90. Let X denote the number of attempts that must be made to gain access to the computer. (a) Find a close form for f(x), the probability distribution of X. f(x) =(0.1) x 1 (0.9) = 1/p , x=1,2,··· =1/0.9=10/9 x (b) Find a close form for F(x), the (0cumulative .1) k 1 (0.9) distribution of X. F(x) = P(X x) = x (0.1) = 0.9 k 1 k 1 k 1 1 0 .1 x = 0.9 1 0.1 1 0.1x = 3 Example I am selling my house, and have decided to accept the first offer exceeding $ K. Assume that offers (money offered to buy the house) are independent random variables with common distribution F, find the expected number of offers received before I sell the house. Solution: N = the number of offers before I sell the house. Find E(N) N has a geometric distribution. N ~g(x, p) X = a price offered by a potential buyer. X has probability distribution F(x). q = P(X ≤ K) = F(K) E(N) = 1/p =1/[1 – F(K)]. 4 Negative Binomial Distribution Consider repeated independent trials that can result in “success” with probability p and a “failure” with probability q = 1 – p. Let X be the number of the trial on which the kth success occurs. X is said to be a negative binomial random variable, and its probability distribution is called the negative binomial distribution with parameters k and p. b*(x; k, p) =P(X = x) E(X) = k/p x 1 k x k = p q k 1 x=k,k+1, … and D(X) = kq/p2 When k=1, negative binomial distribution becomes geometric distribution . 5 Example Toss a coin over and over again until the third head is observed. What is probability that the third head occurs at 5th toss? Solution: (k = 3, x = 5) P(X = 5) = = 5 1 (1 / 2) 3 (1 / 2) 53 3 1 4 (1 / 2)3 (1 / 2) 2 2 6