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assignment_four - Poisson distribution 6 Using...

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Basic Probability Summer 2011 Homework Number Four Some problems on moment Generating functions and the Central Limit Theorem 1) Compute the moment generating function of a binomial random variable with parameters n and p . Use this result to find the mean, variance, and the third moment. If X i is binomial with parameters n i and p i = p for i = 1 , 2 , . . . , n , and the X i are independent, use moment generating functions to show that n i =1 X i is binomial. 2) Use the mgf to show that if X is exponential, so is cX . 3) Find the moment generating function of a geometric random variable and use it to compute the mean and variance. 4) Assuming X N (0 , σ 2 ), use the mgf to show that the odd moments are zero and the even moments are given by μ 2 n = E ( X 2 n ) = (2 n )! σ 2 n 2 n ( n !) . (HINT: use the Taylor series expansion.) 5) Using moment-generating functions, show that as n → ∞ , p 0, and np λ , the binomial distribution with parameters n and p tends to the
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Unformatted text preview: Poisson distribution. 6) Using moment-generating functions, show that as α → ∞ , the gamma distribution Γ( λ, α ), properly standardized, tends to the standard normal distribution. 7) A soft-drink vending machine is set so that the amount of drink dispensed is a random variable with a mean of 200 milliliters, and a standard deviation 1 of 15 milliliters. What is the probability that the average amount dispensed in a random sample of size 36 is at least 204 milliliters? (Hint: use the central limit theorem.) 8) Consider the position of a particle following a random walk: each minute the particle moves north or south by 50 cm, with probability p = 1 / 2. Use the central limit theorem to estimate the probability that the position of the particle will be within 400 cm of the start after 1 hour. 2...
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assignment_four - Poisson distribution 6 Using...

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