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\centerline {\bf Math 563 - Fall 14 - Homework 2} 

\smallskip

\no 1. (from Durrett) Suppose $E Y=0$ and the variance $\sigma^2$ 
of $Y$ is finite. Let $a>0$. Prove
$$
P(Y \ge a) \le \frac{\sigma^2} {\sigma^2 + a^2} 
$$
Hint: apply Chebyshev with $\phi(y)=(y+b)^2$ and optimize 
your result over $b$. 

\medskip

\no 2. One of the hypotheses of the monotone convergence theorem 
is that all the random variables $X_n$ are non-negative.
Prove that the conclusion of the theorem is true without this 
hypothesis if we assume that $E |X_1| < \infty$.


\medskip

\no 3. 
Let $X_n$ be a sequence of RV's which converges to a random
variable $X$ a.s. Let $F$ be a closed subset of $\reals$ which does
not contain the origin. Prove that 
$$\lim_{n \rightarrow \infty} P(|X_n - X| \in F)=0$$
Hint: use continuity of $P$.

\medskip

\no 4. (from Durrett) Let $X_1, X_2, \cdots, X_n$ be RV's
and suppose that the distribution of $(X_1,X_2,\cdots,X_n)$ 
is absolutely continuous with respect to Lebesgue measure on 
$\reals^n$ and the Radon-Nikodym derivative is 
$g_1(x_1) g_2(x_2) \cdots g_n(x_n)$ where the $g_i$ are non-negative
measurable functions but need not have integral $1$. 
Prove $X_1,X_2,\cdots,X_n$ are independent. 

\medskip

\no 5. (from Durrett) Let $\Omega=(0,1)$, $\field$ be the Borel sets
in $(0,1)$, and let $P$ be Lebesgue measure. 
Define RV's by 
\bea
X_n(\omega)=\cases{
0 & if $[2^n \omega]$ is even \cr
1 & if $[2^n \omega]$ is odd \cr}
\eea
where $[x]$ is the largest integer less than or equal to $x$. 
Prove that $\{X_n\}_{n=1}^\infty$ are independent random variables.
Note that this gives a rigorous construction of the probability 
space for flipping a fair coin infinitely many times.

\medskip

{\it The change of variables theorem says if $X$ is a random 
variable and $f$ a real-valued measurable function on the real line 
(with some condition on $f$), then  
\bea
E[f(X)] = \int_\reals \, f(x) \, d\mu_X
\eea
where $\mu_X$ is the distribution of $X$. The point of the next 
two problems is to get more explicit formulae that are amenable to 
computation in the cases that $X$ is discrete or has a density.
}

\medskip

\no 6. Let $X$ be a discrete real-valued random variable. 
Let $x_1,x_2,\cdots$ be its values and let $p_n=P(X=x_n)$. 
Let $g$ be any real valued function on the real line. 
Suppose that 
\bea
\sum_n |g(x_n)| \, p_n < \infty
\eea
Prove that $g(X)$ is a random variable and 
\bea
E[g(X)] = \sum_n g(x_n) p_n
\nonumber 
\eea
Note that I did not say that $g$ was measurable. 

\medskip

\no 7. Let $X$ be a random variable which has a density $f(x)$.
This means its distribution function $F(x)$ satisfies
$F(x)=\int_{-\infty}^x \, f(u) \, du$, i.e., its distribution is 
$f(x)$ times Lebesgue measure. 
Let $g(x)$ be a real-valued measurable function on the real line such that
\bea
\int_{-\infty}^\infty \, |g(x)| \, f(x) \, dx < \infty
\eea
Prove that 
\bea
E[g(X)]= \int_{-\infty}^\infty \, g(x) \, f(x) \, dx
\eea
where the integrals with respect to $dx$  are 
integration with respect to Lebesgue measure.

\medskip

{\it The last three problems are not to be turned in.
They are standard problems in a course like 564. 
I have included them for background. }

\medskip

\no 8. (564 discrete RV's) \, Let $X$ be a discrete RV, $x_1,x_2,\cdots$
its values and $p_n=P(X=x_n)$. The function $f(x)$ which is 
$p_n$ at $x_n$ and is $0$ at points not equal to one of the $x_n$ 
is often called the ``probability mass function'' in 
564 level probability courses. 

Suppose we have a coin with probability $p$ of heads. ($p$ is not
necessarily $1/2$.) The following discrete RV's are of interest:

\no {\bf Binomial:} Fix a positive integer $n$. Flip the coin
$n$ times and let $X$ be the number of heads. So $X$ can be 
$0,1,2,\cdots,n$. 

\no {\bf Geometric:} Flip the coin until you get heads for the first time. 
Let $X$ be the number of tails. (Warning: depending on the author
the definition of $X$ is sometimes taken to be the total number of flips, 
including the final one that gave heads.) So the possible values of
$X$ are the nonnegative integers.

\no {\bf Negative binomial:} Fix an integer $r$. Flip the coin 
until we get heads for the $r$th time. Let $X$ be the total number of tails.
So the possible values of $X$ are the nonnegative integers.

For each of these find the probability mass function and the expected 
value of $X$.

\medskip

\no 9. (564 continuous RV's) \, In a 564 level course, ``continuous 
RV'' usually means that the distribution of the RV is absolutely
continuous with respect to Lebesgue measure and so is given by 
$f(x) dx$. (Having a density implies
the distribution function is continuous, but the converse is very false.)
Here are three common ``continuous'' RV's. 

\no {\bf Uniform:} We say $X$ is uniform on $[a,b]$ if
$f(x) = {1 \over b-a}$ for  $a \le x \le b$ and $f(x)=0$ otherwise.

\no {\bf Exponential:} We say $X$ has an exponential distribution if
$ f(x) = \lambda e^{-\lambda x} $ for $x \ge 0$ and $f(x)=0$ for 
$x<0$. Here $\lambda$ is a positive parameter. 

\no {\bf Normal:} We say $X$ has a normal distribution if 
\bea
f(x) = {1 \over \sqrt{2 \pi \sigma^2}} \exp(- {(x-\mu)^2 \over 2 \sigma^2})
\eea
Here $\mu$ is a real parameter and $\sigma^2$ is a positive parameter. 

Find the expected value of each of these random variables. 

\medskip

\no 10. Let $X$ be a random variable with a normal distribution.
Find the density function of $X^2$. Hint: first find an expression for
the distribution function of $X^2$. 

\end{document}
\end

