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\centerline{\bf University at Albany}

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\centerline{\bf Department of Mathematics and Statistics}

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\centerline{\bf Preliminary Examination}

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\centerline{\bf Probability}

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\centerline{\bf August, 2005}

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Do as many problems as possible!

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1.a. Suppose $X$ is a random variable with
$P(X=1)=P(X=0)=P(X=-1)=1/3$.
Find the characteristic function of $X$.

b. Suppose
$P(X_i=1)=P(X_i=0)=P(X_i=-1)=1/3$ for $i=1,\dots,n$. Also suppose
$X_1,\dots,X_n$ are independent. Let $S_n=\sum\limits_{i=1}^nX_i$. Find
the characteristic function of $S_n$.

c. With the notation of part b and $\sigma^2=Var(X_i)$, find the characteristic
function of $S_n/\sqrt{n\sigma^2}$, and find
(with justification) the limit of this function as $n\rightarrow\infty$.

d. What does the limit in part c tell you? (Give an explanation rather than
a complete proof.)

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2.a. Suppose the events $A_n$ are independent. Prove the following statement.

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$\sum\limits_{n=1}^{\infty}P(A_n)=\infty$, then $P(A_n$ infinitely
often$)=1$.}

b. Suppose that the events $A_n$ are not necessarily independent. Does
the statement to be proved in part a still hold? Justify your response.

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3. Let $X_n$ be a sequence of random variables in some probability space,
and let $a$ be a constant.

a. Define what it means for $X_n$ to converge to $a$ in probability and what
it means for $X_n$ to converge to $a$ almost surely.

b. Give, with justification, an example where $X_n$ satisfies precisely
one of the following two properties:

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i. $X_n$ converges to $0$ in probability.

ii. $X_n$ converges to $0$ almost surely.}

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4. Let $F(x)$ be the distribution function of a random variable.
Prove or disprove the following statement.

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The number of points where $F$ is not continuous is either finite
or countably infinite.}

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5. You play a game as follows:

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A. On the first round, toss a fair coin. If it comes up heads, you lose $1$
point and let $X_1=-1$. Otherwise you win $1$ point and let $X_1=1$.
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B. For the $n$th round (if $n>1$), you do nothing if $X_{n-1}\le 0$ or you did
nothing on the $(n-1)$st round. There you let $X_n=X_{n-1}$. Otherwise
you do nothing with probability $1/3$ and let $X_n=X_{n-1}$. In the remaining
circumstance, you do the following. Toss a fair coin. If it comes up heads,
you lose $1$ point and let $X_n=X_{n-1}-1$. Otherwise you win $1$ point
and let $X_n=X_{n-1}+1$.
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What is $E(X_n)$? Justify your answer. Ideally your answer should mention 
the name of a property which $X_n$ satisfies.

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6. Prove or disprove: Suppose $X_n$ converges in probability to
$X$. Then $E(X_n)\rightarrow E(X)$ as $n\rightarrow\infty$.


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