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\begin{center} \bf
Statistics 930: Probability Theory --- Homework No. 7
\end{center}
\noindent {\sc Instructions.} You should complete the reading of the text through Section 3.5. You should have a clear
understanding of Levy's inversion formula
and Levy's continuity theorem. This is a shorter assignment since we had a test on Monday.
\begin{Problem}
Suppose that for all $x\geq 0$, and suppose the random variable $X$ satisfies the bound
$$
P(X \geq A + B \sqrt{x}) \leq e^{-x},
$$
where $A$ and $B$ are nonnegative constants and $n$ is an integer. Show that one has
$$
E[X] \leq A + \frac{B \sqrt{\pi}}{2}.
$$
Hint: It may be useful to note that $X \leq A +(X-A)_+$, and you may need to recall the value of $\Gamma(1/2)$.
\end{Problem}
\begin{Problem}
\begin{itemize}
\item Remember, compute, or look up the characteristic function of the Gamma density. Suppose that $X_1$ and $X_2$ are independent and identically distributed
and suppose that $X_1+X_2$ has the exponential distribution with mean one. Write down an explicit formula for the density of $X_1$. Marvel at
the beauty of this.
\item If $Y_1$ and $Y_2$ are independent, normal, mean zero, variance one, find the distribution of
$$
R=\sqrt{Y_1^2 + Y_2^2}
$$
\end{itemize}
\end{Problem}
\begin{Problem}
Show that there does not exist a characteristic function $\phi(t)$ such that the derivative $\phi'(t)$ exists for all $t$ and such that
$\phi'(t)$ is also a characteristic function. Note: This would be super easy if we were to assume
that the distribution associated with $\phi(t)$ had a finite first moment. Unfortunately we know that it is possible for $\phi'(t)$ to exist
with out the first moment existing. You need an argument where you do not assume that you have a first moment.
\end{Problem}
\begin{Problem}
Consider the sum
$$
A_n(t)=\frac{1}{2n+1} \sum_{k=-n}^n e^{itk/n}.
$$
\begin{itemize}
\item This is the characteristic function of a random variable $Z_n$. Describe $Z_n$ as the result of an experiment.
\item
For large $n$ the discrete random variable $Z_n$ looks a lot like a random variable that has a density. Describe that density.
\item
Now, using the theory of characteristic functions, explain why it is obvious that one has
$$
\lim_{n\rightarrow \infty} \frac{1}{2n+1} \sum_{k=-n}^n e^{itk/n} = \frac{\sin t}{t}.
$$
\item Now explain why the last formula is actually obvious from the definition of the Riemann integral.
\end{itemize}
A useful lesson on from this exercise is that the
integral of the RHS over $(-\infty, \infty)$ is $\pi$ but the integral of the LHS over this
interval is (pretty much) non-sense. This observation gives us a reminder that it is fine to {\bf calculate boldly},
but one still has to avoid drifting off into silliness.
\end{Problem}
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