Let X denote the reaction time, in seconds, to a certain stimulant and Y denote the temperature (°F) at which a certain reaction starts to take place. Suppose that two random variables X and Y have the joint density 0 < x< 1; 0 < y<1 0, (4xy, f(x, y) = {* elsewhere a.) P(0 s x s and srs); b.) Find P(X

Let (X, Y) be a pair of jointly continuous random variables with joint p.d.f. 3 f (x, y) : =, 0sx< 1, 2²< y< 1. 2' (a) Find the marginal p.d.f. fx(x) of X and the marginal p.d.f. fy (y) of Y. (Ь) Find the covariance Cov(X, Y) of X and Y. (c) Find the conditional p.d.f. fy|x(y|x) of Y given X = x, for any x E (0, 1). (d) Find the conditional expectation E(Y|X = 1/2) of Y given X = 1/2. 2.

Q. 3 Let continuous random variables X and Y be independent identically distributed random variables with the following respective pdfs: fx(x) = 2e-2x;x 2 0 and fy (y) = 2e-2y;y > 0 Define a new random variable Z = X + Y. Derive moment generating function of Z, i.e. Mz(t).

The continuous random variable X on [-1,1] has pdf given by fx(x) = k(3x + 5) for x = [1,1] where k is a constant that you are going to determine. fx(x) = 0 outside the interval [-1,1] Hint: consider Calculate the following: i. k ii. E(X) iii. Var(X)