Suppose that we define a distribution in the following way. Let X have the following PDF:...

  

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Suppose that we define a distribution in the following way. Let X have the following PDF: 1+ ax f(x; a)= -1≤x≤1, -1≤a ≤ 1. 2 We will assume that X₁,..., X, are i.i.d. samples from f(x; a) for a fixed a. 1. What is the expectation of 3.X? 2. What is the variance of 3X? " 3. Using the central limit theorem, what is the normal approximation for the distribu- tion of 3X? (Hint: if √n(X₂-μ) N(0,02), then the normal approximation for X is N(μ, o²/n)) 4. Suppose that n = 50 and the true parameter is ao = 1/3. What is Pr(3X, col 1/6) using the central limit theorem approximation? Suppose that we define a distribution in the following way. Let X have the following PDF: 1+ ax f(x; a)= -1≤x≤1, -1≤a ≤ 1. 2 We will assume that X₁,..., X, are i.i.d. samples from f(x; a) for a fixed a. 1. What is the expectation of 3.X? 2. What is the variance of 3X? " 3. Using the central limit theorem, what is the normal approximation for the distribu- tion of 3X? (Hint: if √n(X₂-μ) N(0,02), then the normal approximation for X is N(μ, o²/n)) 4. Suppose that n = 50 and the true parameter is ao = 1/3. What is Pr(3X, col 1/6) using the central limit theorem approximation?