(a) Let X be a random variable with PDF ƒ(x) = 1−|x|, |x| ≤ 1. Obtain the MGF of X. (b) Find the PDF of a random variable Y with MGF T 2 My(t) = 1 = ( ³² = ¹) ²³₁ t

x > 1 A RV X has the PDF fx(x) = otherwise (a) Find the MGF Mx(t) of X and clearly define the values of t for which it exists. (b) Find the mean (µ) and variance (o?) of the RV X using the MGF from (a).

1. Fix r € N, and suppose that Y₁,..., Yn are independent and identically distributed NegBin(r, ) random variables, each with probability mass function given by ¡v (u | 0) – (1 + r − ¹) (1 - fy y (1 - 0) ³0⁰ for y ¤ N and unknown 0 € (0, 1). Here the number of successes r in the negative binomial is known.