MTH 504 Homework

Back

Homework 1

Let $\{X_i\}_{i=1}^{\infty}$ be a collection of iid random variables. If $f$ is a tail-measurable function, show that $f$ is a constant.
a. If $X$ is a random variable with continuous distribution and $B$ is such that $\Prob(B) > 0$ , show that
$\E[ X | B ] = \frac{1}{\Prob(B)} \int X 1_B d\Prob$
b. (From textbook) Let $(X,Y)$ be an absolutely continuous random varaible with piecewise-continuous density $f(x,y)$ . Prove that for every non-negative measurable $h \colon \R \to \R$ ,
$E[ h(X) | Y ] = \frac{ \int_{-\infty}^{\infty} h(x) f(x,Y) dx}{ \int_{-\infty}^{\infty} f(x,Y) dx} \quad \almostsurely$
Even though $\Prob(Y = y) = 0$ for all $y \in \R$ , use the preceding to justify the classical definition,
$P(X \leq a | Y = y) = \frac{\int_{-\infty}^a f(x,y)dx}{\int_{-\infty}^{\infty} f(x,y)dx} \quad \forall a \in \R$
a. If $X$ is a martingale and $\psi$ is convex, then $\psi(X)$ is a submartingale if $\psi(X_n) \in L^1(\Prob)$ for all $n$ . b. If $X$ is a submartingale, then $\psi(X)$ is a submartingale if $\psi$ is nondecreasing in addition to convex and integrable.
Suppose $X_m$ is a submartingale bounded in $L^1$ . Show that
$\lim_{m \to \infty} E[X_m] = \sup_m \E[ X_m ]$