Suppose that a pdf for a continuous random variable Y takes the form 1 fV) = e-V-H)lo o (1+e-6-p)lo)2 y E R where μ Ε R, σ >0 If the two parameters H and take the values -4.5 and 1.4 respectively, compute the probability P(Y > -1.1).

Let Z be a discrete random variable withE(Z)=0. Does it necessarily follow that E(Z3)=0? If yes, give a proof; if no, give a counterexample.

Exercise 3. Let X be a random variable with mean µ and variance o². For a € R, consider the expectation E((X − a)²). a) Write E((X - a)²) in terms of a, μ and σ². b) For which value a is E((X − a)²) minimal? c) For the value a from part (b), what is E((X − a)²)?

Let a, µ E R. Calculate f exp{-ax²/2+µx}dx. (Hint: recall that the PDF for a normal random variable Z ~ N(H, o2) is p(2) = (270²)-1/2 exp{-(z – µ)²/(20²)} and that p(z)dz = 1.)