Homework 11 - ¯ X n is the sample mean that is ¯ X n = 1...

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550.420 Introduction to Probability Homework 11 Due November 17, 2011 1. (2 points) Suppose that there are 25 students in a probability class. What is the expected number of birthdays that belong to only one student? 2. (2 points) Thieves stole four animals at random from a farm that had seven sheep, eight goats, and Fve burros. Calculate the covariance of the number of sheep and the number of goats taken. 3. (2 points) ±or a random variable X , the moment generating function M X ( t ) = p 1 81 P ± e t + 2 ² 4 . ±ind P [ X 2]. 4. (2 points) A random variable X has the moment-generating function M X ( t ) = 1 6 e - t + 1 2 e - 2 t + 1 3 e t . ±ind the mean and variance of X . 5. (2 points) Let X be a random variable with Fnite variance. Suppose that you don’t know the variance and want to estimate it. You take a random sample, X 1 , X 2 , . . . , X n of independent observations from the distribution of X and set S 2 n = 1 n - 1 n s j =1 ( X i - ¯ X n ) 2 , 1
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Unformatted text preview: ¯ X n is the sample mean, that is, ¯ X n = 1 n ∑ n i =1 X i . Show that the random variable S 2 n , which is called the sample variance, has expectation σ 2 . In statistical terminology, S 2 n is said to be an unbiased estimator of σ 2 . Extra Credit (3 points) 6. (Continuation of problem 5) Using the notation and assumptions of prob-lem [5], assume further that the third central moment of the distribution of X equals zero. Prove that the sample mean and the sample variance are uncorrelated, using the following steps: (a) Let Y j = X j-μ X for 1 ≤ j ≤ n , and let T 2 n denote the sample variance of the Y j ’s. Show that Cov ( ¯ X n , S 2 n ) = Cov ( ¯ Y n , T 2 n ) . (b) Prove that Cov ( ¯ Y n , T 2 n ) = 0 . (c) Conclude that ¯ X n and S 2 n are uncorrelated. 2...
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Homework 11 - ¯ X n is the sample mean that is ¯ X n = 1...

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