ALL ABOUT PROBABILITY

Show all necessary solutions. 1. A random variable X denotes the number of books bought by a customer in a bookstore. We know that the random variable has the distribution P(0 3. a. Under which conditions on a, b, c, d is the function F a cumulative distribution function? b. Assume that F is the c.d.f. of an r.v. X. Compute the law of X. 3. Let X ~ Geometric(p) with pe (0, 1). Prove that for every a, be N we have P(X = a +b|X > a) = P(X =b). 4. Let X be a discrete random variable with the following distribution: P(X = -3) =a, P(X =-2) = 3, P(X =-1) =b, P(X = 0) = 16, P(X = 1) = c, P(X = 2) = 32 a. Find a, b, c. b. Calculate E(3+ 2X), E(3+2X)', Var(3 + 2X). 5. Derive the expected value of the random variable X with distribution Negative Binomial(r, p)