STOCHASTIC

(a) Show that Xi has less probability mass in the "middle" and more mass in the "tails"
than X; that is, A"i is more spread out than X.
(b) Show that EXt = EX.
Let F and Fi be the cumulative distribution functions of X and Xit respectively. Define
(c) Show that 7\ (x) £ 0 for all x e [a,b], and 7\ (6) = 0.
(d) Construct a discrete random variable Zt such that P[Xi g x] = Pt-Y+Z! g x] Vx,
with£[Zi|A"] = 0.
The random variable y is said to differ from A 1 by a sequence of MPSs if there exists a finite
sequence of random variables
X = X0, Xi, X2,..., XN, Ajv +1 = Y
such that X,+i differs from Xt by a MPS. Let G be the cumulative distribution function of Y;
assume that EX = EY, and
I"* [