Problem 1. Study of a density function and a random variable. (3 pts) Define a function f on R by f(x) = e if x < 0 and 0 otherwise. 1. (0.5 pt) Prove that f is the density function for a certain random variable, which we denote by X. (Hint: calculate the integral.) 2. (0.5 pt) Determine the distribution function of X. (Hint: calculate the integral..) 3. (0.5 pt) Calculate the expectation of X. (Hint: calculate the integral...) 4. (1.5 pts) Let Y = 2X+1. (a) (0.5 pt) Determine the distribution function of Y. (Hint: Y