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Introduction to Statistical Inference 1. Wackerly, 4.130 Prove that the variance of a beta-distributed random variable with parameters a and 8 is 2 a8 2. Wackerly, 5.100 Let Z be a standard normal random variable and let Y1 = Z and Y2 =Z². (a) What are E(Y1) and E(Y2)? (b) What is E(Y1Y2)? [Hint: E(Y1Y2) = E(Z*).] (c) What is Cou(Ys,Y2)? (d) Show that Y/ and Y2 are dependent. 3. Wackerly, 6.7 Suppose that Z has a standard normal distribution. (a) Find the density function of U = Z?. (b) Does U have a gamma distribution? What are the values of a and ,8? (c) What is another name for the distribution of U? 4. Wackerly, 6.15 Let Y have a distribution function given by Find a transformation G(U) such that, if U has a uniform distribution on the interval (0,1) G(U) has the same distribution as Y. 5. Let Y/ and Y2 be independent and uniformly distributed random variables on the interval [0,0]. Find the (a) probability density function of U = min(Y1,Y2) and U2 = max(Y1,Y2). (b) mean and variance of U1. (c) E(U2 - U1). Does it equal to E(U/)?

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