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# tutorial5 - THE UNIVERSITY OF HONG KONG DEPARTMENT OF...

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Unformatted text preview: THE UNIVERSITY OF HONG KONG DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE STAT 1301 PROBABILITY AND STATISTICS I EXAMPLE CLASS 5 Discrete Distribution Discrete Uniform Distribution A random variable X has a discrete uniform[1, N] distribution if P , 1, 2, , , where N is a natural number. Bernoulli Distribution A Bernoulli random variable X takes on only two values: 0 and 1, with probabilities 1 and p, respectively. Its pmf is given by P 1 , 0, 1. Binomial Distribution Suppose that n independent trials, each of which results in a “success” with probability p, are to be performed. If X represents the number of successes that occur in the n trials, then X is said to be a binomial random 1 , 0, 1, , . variable with parameters (n, p). Its pmf is given by P C Poisson Distribution A random variable X that takes on one of the values 0, 1, 2 … is said to be a Poisson random variable with parameter , 0, if its pmf is given by P , 0, 1, . ! Geometric Distribution Consider independent trials, each of which is a success with probability p. If X represents the number of the first trial that is a success. Its pmf is given by P 1 , 1, 2, . Negative Binomial Distribution A random variable X which denotes the number of trials needed to amass a total of r successes when each trial is independently a success with probability p, then X is said to be a negative binomial random variable 1 , , 1, . with parameters p and r. Its pmf is given by P C Hypergeometric Distribution If there are 2 types of objects which N are of type I and M are of type II. If a sample of size of n is randomly chosen, then X, the number of type I selected, has pmf P Ex.1 CC C , max ,0 min , N . Find E(X) if X is a random variable with discrete uniform distribution [A, B], where A and B are . (Hint: ∑ natural numbers with A < B. Hence, show that Var(X) ) Ex.2 A typesetter, on the average, makes one error in every 500 words typeset. A typical page contains 300 words. What is the probability that there will be no more than two errors in five pages? Ex.3 A newsboy purchases papers at 5 dollars and sells them at 6.5 dollars. However, he is not allowed to return unsold papers. If the daily demand is a binomial random variable with n = 8 and p = 0.5, approximately how many papers should he purchase so as to maximize his expected profit? Ex.4 An auto parts store has 200 rebuilt starters in stock, of which 4 are defective. In response to a purchase order, they randomly select 3 starters from their stock. What is the probability that the customer is given 2 defective starters? Ex.5 Let g(x) be a function with ∞ (a) if ~Poisson , then E (b) if ~negative binomial , X E and ∞ and ∞ 1. , then E 1 1 ∞. Prove that X E 1 . ...
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