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theory of probability 3

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MATH 464 HOMEWORK 4 SPRING 2013 The following assignment is to be turned in on Thursday, February 14, 2013. 1. Let X be a discrete random variable on a probability space (;F; P). Let g : R ! R be a function and set Y = g(X), i.e. Y : ! R is dened by Y (!) = g(X(!)) for all ! 2 : Prove that Y is a discrete random variable. 2. Let 0 0. Compute the following: a) P(2 X 4) b) P(X 5) c) P(X is even) give each answer in exact form and, with the choice of = 2, give a decimal approximation to the above which is accurate to 3 decimal places. 4. Let X be a discrete random variable whose range is f0; 1; 2; 3; g. Prove that E(X) = 1Xk=0 P(X > k) : 5. Compute the expected value of the geometric random variable with pa- rameter 0 0 an integer. For any 0 k n, denote by Pk = P(X = k). Compute 1
2 SPRING 2013 the ratio Pk??1 Pk for 1 k n : Show that this ratio is less than one if and only if k 0. Let g : R ! R be the function g(x) = x(x ?? 1). Set Y = g(X). Find E(Y ). 8. Let X be a function whose range is f1; 2; 3; g. Consider the values P(X = n) = 1 n(n + 1) for any n 1 : Does this function X dene a discrete random variable? If so, what is E(X)?

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