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\centerline {\bf Math 563 - Fall 15 - Homework 4} 

\smallskip

\no 1. 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 density can be written in the form
$g_1(x_1) g_2(x_2) \cdots g_n(x_n)$ where the $g_i$ are non-negative
measurable functions but the integral of $g_i$ over $\reals$
need not be $1$. 

\smallskip
\no (a) Prove that each $X_i$ is an absolutely continuous
random variable. 

\smallskip
\no (b) Prove $X_1,X_2,\cdots,X_n$ are independent. 

\medskip

\no 2. Let $X$ be a discrete real-valued random variable, i.e., it has 
countable range.  
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 3. (Resnick) 
Let $X_n$ be a sequence of {\bf independent} random variables. 
Consider the random
power series $\sum_{n=1}^\infty X_n z^n$. 
Its radius of convergence $R$ is a random variable. 
Prove that $R$ is a constant almost surely. 
In other words, prove there is a constant $c$  such that 
$P(R=c)=1$. 
Hint: the radius of convergence $R$ is 
given by $R^{-1}= \limsup_n |X_n|^{1/n}$.  
Note that $R$ can be $\infty$ or $0$.  

\medskip

\no 4. (a) Let $X$ and $Y$ be independent random variables which 
take values in the integers. Prove that the distribution of $X+Y$ 
is given by 
$$
P(X+Y=n) = \sum_{m= -\infty}^\infty P(X=m) P(Y=n-m)
$$

\smallskip

\no (b) $X$ has a Poisson distribution with parameter $\lambda>0$
if it takes on the values $0,1,2,\cdots$ and 
$$
P(X=n)= \frac{e^{-\lambda} \lambda^n}{n!}
$$
Show that if $X$ and $Y$ are independent random variables, $X$ has 
a Poisson distribution with parameter $\lambda$ and 
$Y$ has a Poisson distribution with parameter $\mu$, then $X+Y$
has a Poisson distribution. What is the parameter for $X+Y$ ?

\medskip

\no 5. (Resnick) Let $X$ be a random variable. 
A number $m$ is called a median of $X$ if 
$P(X \ge m) \ge 1/2$ and $P(X \le m) \ge 1/2$. 
You should convince yourself that there is always at least one median, 
but it need not be unique. 
Recall that the mean of $X$ is $\mu=E[X]$ if $X$ is integrable. 

\smallskip
\no (a) Prove that if $X$ has finite variance, then 
the minimum of \\ 
$f(a) = E[ |X-a|^2]$ occurs at $a=\mu$, the mean. 

\smallskip

\no (b) Let $m$ be a median of $X$. 
Prove that the minimum of $g(a) = E[ |X-a|]$ occurs at $m$.

\end{document}
\end
\no 5. (Resnick) 
Let $X_n$ be a sequence of random variables. Show they are 
independent if and only if for $n \ge 2$, the two $\sigma$ fields 
$\sigma(X_1,X_2, \cdots, X_{n-1})$ and $\sigma(X_n)$ are independent. 
(easy induction) 

\medskip

