Suppose X follows an exponential distribution with parameter A, then P(X ≤ 1|X ≤ 2) = e-^. (b) For a random variable X, suppose P(X>0) > P(X <0). True or False: The expectation of X must be positive. (c) Let ₁ and ₂ be two estimators of a parameter 0, where ê, has a strictly higher variance than 2. True or False: If the MSE criterion chooses the estimator ₁, then 2 must be biased.

Suppose that X and Y are unknowns with E ( X ) = 10, Var ( X ) =4,  E ( Y ) = 12  and Var ( Y ) = 100.  In addition,  suppose that the correlation coefficient of X with Y is .8.  Then what is the expected value of XY?

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

Let X₁, X2, X3, Xn be a random sample with unknown mean EX; = µ, and unknown variance Var(X₂) = o². Suppose that we would like to estimate 0 = μ². We define the estimator as 2 • - (™)² - [ 2x]* Xk to estimate 0. Is an unbiased estimator of ? Why?