PROBLEMS 2.1 Apply the inverse-transform method to generate a random variable from the discrete uniform distribution with pdf = { n+1' 0 x = 0, 1,..., n, otherwise.

4. The standard deviation of X, denoted SD(X), is given by SD(X) = Var(X). Find SD(aX + b) if X has variance of o?.

Question 1 1. Suppose you are given that a continuous random variable has a moment generating function of 1 M (t) = 1-3.2t Find the mean and variance of the random variable. mean = var =

Question 5 Suppose X is a normal random variable with mean and variance o². (i) If a and b are constants, show that aX + b has a normal distribution and find its mean and variance. (ii) If Z = (X − μ)/o, deduce that Z~ N(0, 1). (iii) Using (ii) find P(X 0 is a constant, show that P(µ – co < X < μ+co) does not depend on μ or o.