1) Assume a relationship Y = g2(91(X)) between the two random variables X and Y where gi and g2 are two functions. To compute the mean of Y, we can choose any of the following equivalent expressions: E(Y) = | fr (u)ydy = | fn(x)(u)g2(u)du = / fx(#)g2(9n(x)dr (1) -00 where fy (y), f1(x)(u) and fx(x) are the probability density functions (p.d.f.s) of Y, 91(X) and X, respectively. Now consider Y = X? where X is N(0, o2). a) Determine the p.d.f. fy (y) of Y. Hint: for y > 0, fy (y) Prob(Y < y) = Prob(-< X < VI) = % x (x)dx. b) Then evaluate explicitly E(Y)= S fy(y)ydy.

The random variables X,Y have variance Var(X)=36 and Var(Y)=1 and their correlation is Cor(X,Y)=−34. Calculate Var(X+Y) with a full explanation

Let X be a random variable with mean E[X] = 20 and variance Var(X) = 3. Define Y = 3 – 6X. Calculate the mean and variance of Y.

Q. Let the joint pmf of x and y be defined by f (x,y) = (x+y)/32  where x = 1,2 and y = 1,2,3,4

If E(X) = 6 and E[(X)(X-1)] = 51,  a) find the second moment  b) find the variance of X Please provide your working and round your answers to 3 decimal places

Prove that the variance of b, V(b) = o²(X'X)−¹

= 10. We do Let X and Y be statistically independent random variables with Var(X) = 4 and Var(Y) not know E[X] or E[Y]. Let Z = 2X +9Y. Calculate the correlation coefficient defined as p(X, Z) = Cov(X,Z) √Var(X)Var(Z)

X (t) is a random process having mean = 2 and auto correlation function. Rxx (7) = 4 [e-0.2 ltl + 1. Let Y and Z be the random variables obtained by sampling X (t) att = 2 and t = 4 respectively. Find the variance of the random variable W = Y -Z.