Discuss the below:
Q1. Let the random variable X have the pdf
f(x) = 2/sqrt(2(pi)) e^(-x^2/(2) ), 0
Find the mean and the variance of X.
Hint: Compute E(X) directly and E(X^2) by comparing the integral with the integral representing the variance of a random variable that is N(0,1).
Q2. Compute P(X_1 + 2X_2 - 2X_3 > 7), if X_1, X_2, X_3 are iid with common distribution N(1,4).
Q3. Let X and Y have a bivariate normal distribution with parameters u1 = 20, u2 = 40, sigma1^2 = 9, sigma2^2 = 4, and p = 0.6. Find the shortest interval for which 0.90 is the conditional probability that Y is in the interval, given that X = 22.