Let Z be the standard normal random variable. (a) Find the value of the probability density function fx (x) of X = |Z + 11 for x = 1. (b) Find the value of the probability density function fy(y) of Y = e² for y = 2. (c) Find E[Y4] (Hint: this problem is closely related to the moment generating function for the normal distribution.)

Suppose that X and Y have a joint probability density function given by Sce-32-5y if r, y 2 0 fx.x(r, y) = otherwise (a) Determine the value of the normalization constant c. (b) Find the marginal probability density function fx and state the name of the distribution of X. (c) Find the conditional probability density function fy|x=z•

Let X be a random variable with the probability density function f(x) = { ae; x>0 0 ; otherwise Find (i) Value of a

Consider the following probability density function of a random variable X f(x) c(x+1)², if −1≤ x ≤ 1, {ox 0, otherwise. (a) What is the value of c such that f (x) is a valid probability density function? (b) What is the mean and variance of X? (c) Let Y = X2, what is the mean of Y? =