# Tag Archives: JCM_math545_HW1_S14

## Simple Numerical Exercise

Let $$\omega_i$$ , $$i=1,\cdots$$ be a collection of mutually independent, uniform on $$[0,1]$$ random variables. Define

$\eta_i(\omega)= \omega_i -\frac12$

and

$X_n(\omega) = \sum_{i=1}^n \eta_i(\omega)\,.$

1. What is $$\mathbf{E}\,X_n$$ ?
2. What is $$\mathrm{Var}(X_n)$$ ?
3. What is $$\mathbf{E}\,X_{n+k} | X_n$$ for $$n, k >0$$ ?
4. What is $$\mathbf{E}(\,X_5^2 \,|\, X_3)$$ ?
5. [optional] Write a computer program to simulate some realizations of this process viewing $$n$$ as time. Plot some plots of $$n$$ vs $$X_n$$.
6. [optional] How do you simulations agree with the first two parts ?

## First Borel-Cantelli Lemma

The first Borel-Cantelli lemma is the principle means by which information about expectations can be converted into almost sure information.

Let $$(\Omega, {\cal F}, P)$$ be a probability space and let $$A_k$$, $$k=1,2,3,…$$ by a sequence of events in $${\cal F}$$ for which
$\sum_{k=1}^{\infty} P(A_k) < \infty$
Prove the (first) Borell-Cantelli lemma which says that
$P\Big\{\bigcap_{N=1}^{\infty}\bigcup_{k=N}^{\infty} A_k \Big\}=0$
Interpret this event whose probability is zero. (Say sometime like “it is the event in which all of the events $$A_k$$ happen”. This of course not the right answer.)
Now let us use this result: Let $$X_k$$ be a sequence of random variables and suppose that there is another random variable $$X$$ such that if
$A_k =\{\omega\in\Omega ~|~ ~~ |X_k(\omega)-X(\omega)|>\delta \}$
then
$\sum_{k=1}^{\infty} P(A_k) < \infty$
for all $$\delta >0$$. Show that $$X_n \to X$$ with probability one. (That is the set of $$\omega$$ so that $$X_n(\omega) \to X(\omega)$$ has measure (probability) one.)

## Chebychev Inequality

Let $$X$$ be a real valued random variable. Prove the Chebychev inequality:
$\mathbf{P}\{|X| \geq \lambda\} \leq \frac{\mathbf{E}\{|X|^p \}}{\lambda^p }$
for all $$1\leq p <\infty$$ as well as its exponential form
$\mathbf{P}\{|X| \geq \lambda\} \leq \frac{\mathbf{E}\{e^{k|X|} \}}{e^{k\lambda} }$
when the integrals on the right are finite. Here $$k$$ and $$\lambda$$ are positive constants.

Hint: combine the facts that $\mathbf{P}\{|X|^p \geq \lambda\} = \mathbf{E} \mathbf{1}_{\{|X|^p \geq \lambda\}}$

and

$\mathbf{1}_{\{|X|^p \geq \lambda\}}\lambda^p \leq \mathbf{1}_{\{|X|^p \geq \lambda\}}|X|^p$ and $1= \mathbf{1}_{\{|X|^p \geq \lambda\} }+ \mathbf{1}_{\{|X|^p <\lambda\}}\, .$