Throughout the following problems, let φ and Φ be the probability density function and cumulative distribution function, respectively, of the standard normal distribution. Suppose Z is a standard normal random variable and let X = 3Z + 1.
(a) Express P(X ≤ x) using the function Φ.
(b) Differentiate the expression from (a) with respect to x to get the density function of X, f(x), and express the result in terms of φ. [Hint: remember that Φ'(z) = φ(z) and don’t forget the chain rule]
(c) Find P(-1 ≤ X ≤ 1) [hint: use the relationship between X and Z].
*Use the standard normal distribution table.