Let X1,...,Xn be a random sample from a distribution with mean μ and variance σ2. Find the value of the following quantities (as a function of μ and σ2). (d) Var(4X1 −3X2 +2X3 −X4) (e) E{(X1 − X2)2} (f) {E(X1 − X2)}2

f (x) = 4xc for 0 ≤ x ≤ 1 ; Find the constant c so that f (x) is apdf of some random variable X, and then find the cdf, F (x) = P(X ≤ x). Sketch graphsof the pdf f (x) and the cdf F (x), and find the mean μ and variance σ2.

(a) Let Y be a random variable distributed as X. Determine E(Y) in terms of r. (b) Let {X1, X2, . .. , Xn} be a random sample drawn from a normal distirbution with mean u and 1 variance o?. Denote S E-(X; – X)² as the sample standard deviation. Use the 1 n - result in part (a), or otherwise, to find E(S). (c) Find an unbiased estimator for the population standard deviation o.

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?

f(X)= 3/8( 4x - 2x2 )         0<* x <* 2     (* less or equal to) a.  Find the variance of X.

If E[X] = 1 and Var(X) = 5, use definition of variance and properties of expectation to find (a) E[(2 + X)^2] (b) V ar(aX) for any constant a. (c) V ar(X + b) for any constant b. (d) V ar(4 + 3X)

Let M₁ (2) and M₂(z) be the m.g.f. of r.v.'s X₁ and X2, respectively. Suppose that M (0) = M₂(0) and M"(0) > M"(0). Which of the following statements is true? The mean of X₁ is greater than the mean of X₂ The variance of X₁ is equal to the variance of X₂. The variance of X₁ is greater than the variance of X₂. The variance of X₁ is less than the variance of X₂