2.3.3. Let f(¤1, 2) = 21rf, 0 < 01 < ¢2 < 1, zero elsewhere, be the joint pdf of X1 and X2. (a) Find the conditional mean and variance of X1, given X2 = x2, 0 < T2 < 1. (b) Find the distribution of Y = E(X1|X2). (c) Determine E(Y) and Var(Y) and compare these to E(X1) and Var(X1), re- spectively.

Let Y, represent the ith normal population with unknown mean #, and unknown variance of for i=1,2. Consider independent random samples, Ya, Ya, Yin, of size n,, from the ith population with sample mean Y, and sample variance S² = (₁-P) ². (d) Define V₁, a function of S1, that has a chi-square distribution. (e) Is V₁ a pivotal function to find a confidence interval for o?? Explain with argument. (f) Find a (1-a) x 100% confidence interval for of

f(x) = (2x + 1)/c, 0 ≤ x ≤ 4. Find the variance of the distribution. 438/c² 409.2/c² O 431.6/c² O 419.6/c²