# HW6 - 9 that the class average would be within 5 of 75...

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OR 3500/5500, Summer’09, Chen Homework 6 Due on Wednesday, June 10, 3:00pm. Problem 1 Let X be a continuous random variable with a density f X ( x ) = λ 2 e - λ | x | for -∞ < x < . Here λ > 0 is a parameter. (a) Compute the moment generating function of X . What is the range on which the moment generating function is deﬁned? (b) Use the moment generating function from part ( a ) to compute the mean, the second moment and the variance of X . Problem 2 From past experience, a professor knows that the test score of a student taking her ﬁnal examination is a random variable with mean 75. (a) Based only on this information , give an upper bound to the probability that a student’s test score will exceed 85. Suppose now that in addition the professor knows that the variance of a student’s test score is equal to 25. (b) What can be said about the probability that a student will score between 65 and 85? (c) How many students would have to take the examination so as to ensure, with probability at least
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Unformatted text preview: . 9, that the class average would be within 5 of 75? Problem 3 Suppose now this professor believes that the test score of a student taking her ﬁnal examination is a NORMAL distributed random variable with the same mean 75 and the same variance 25. (a) Compute the probability that a student’s test score will exceed 85. (b) Compute the probability that a student will score between 65 and 85? (c) How many students would have to take the examination so as to ensure, with probability at least . 9, that the class average would be within 5 of 75? Compare those with the results in problem 2. The following problem is optional for 3500 students, mandatory for students enrolled in 5500 Problem 4 If X is a random variable with moment generating function φ X , show that (a) φ X (2 t ) ≥ φ X ( t ) 2 for all t (b)For a constant a > 0, P [ | X | ≥ a ] ≤ e-ta φ | X | ( t ) , ∀ t > . 1...
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