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\centerline {\bf Math 563 - Fall '21 - Final (take home)} 

\medskip

\no {\bf Due} : You can work on the final for one week (7 consecutive days).
It is due at the end of that week, but no later than
Wed, Dec 15 at 11:59pm. Upload your solutions to gradescope.

\medskip

\no {\bf The fine print: } You are supposed to do this exam on your
own. This means you should not talk to anyone about the exam.
You can ask me questions, but I will only answer questions about
what the problem is asking, not about how to do it. 
You can consult your class notes, past homeworks and Durrett.
You cannot consult any other sources including the web. 

\medskip

\centerline{\bf Do 5 of the 6 problems. Do not turn in 6 problems.}

\medskip

\no 1. Let $X_n$ be an i.i.d. sequence of random variables. 
Each $X_n$ has an exponential distribution with parameter $\lambda$.
So its density is $\lambda \exp(-\lambda x)$ for $x \ge 0$. 
Prove that 
\bea
\limsup_{n \ra \infty} \frac{X_n}{\ln n} = \frac{1}{\lambda} \, \, a.s. 
\eea
Hint: Given a $c>0$, what can you say about 
$P(\frac{X_n}{\ln n} \ge c \, \, i.o.)$. 

\bigskip


\no 2. Let $A_n$ be a sequence of independent events such that 
$\lim_{n \ra \infty} P(A_n)$ exists and is not $0$ or $1$. Prove that 
\bea
\frac{1}{\sqrt{n}} \sum_{k=1}^n [1_{A_k} -P(A_k)]
\eea
converges in distribution to a normal random variable and 
find the mean and variance of this limiting normal random 
variable. 

\bigskip

\no 3. Let $X_n$ and $X$ be random variables on the same probability
space. Assume that $E[|X_n|]<\infty$  and $E[|X|]<\infty$.

Here are four types of convergence: 
(1) pointwise convergence almost surely, (2) convergence in probability,
(3) convergence in distribution, and (4) convergence in $L^1$, i.e.,
$E |X_n-X|$ converges to zero.

\smallskip
\no (a) For each of the four types of convergence 
prove or disprove that if $X_n$ converges to $X$ then $E[X_n]$ 
converges to $E[X]$. 

\smallskip
\no (b) For each of the four types of convergence 
prove or disprove that if $X_n$ converges to $X$ then $E[\cos(X_n)]$ 
converges to $E[\cos(X)]$.

\newpage

\no 4. A random variable $X$ has a geometric distribution 
if it takes values in the non-negative
integers and $P(X=k)=(1-p)^k p, k=0,1,2,\cdots$. Here $p$ is a parameter
between $0$ and $1$.
Suppose you have a coin with probability of heads equal to $p$. Flip it until
you get heads for the first time. Let $X$ be the total number of flips.
Then $X$ has a geometric distribution.

Let $\alpha>0$. 
Let $X_n$ be a sequence of random variables such that 
$X_n$ has the geometric distribution with $p=\alpha/n$. 
Let $Y_n = X_n/n$. Prove that $Y_n$ converges in distribution and 
find the limiting distribution. A list of some facts about common discrete and
continuous distributions is on the next page and can be used for this problem. 
Note that it gives the moment generating function $M(t)=E[\exp(tX)]$ rather
than the characteristic function.

\bigskip

\no 5. Let $\xi_n$ be i.i.d. RV's. We assume they are integrable. 
Let $X_n = \sum_{k=1}^n \xi_k$. 
Assume that $E[\xi_n] \neq 0$. Recall that
this implies $X_n$ is not a martingale. 
Define $Z_n=\exp(\alpha X_n)$ where $\alpha$ is a constant.

\smallskip
\no (a) Show that if there is an $\alpha$ such that $\exp(\alpha \xi_n)$
is integrable and 
$$E[\exp(\alpha \xi_n)]=1$$
then $Z_n$ is a martingale.

\smallskip
\no (b) Now suppose $\xi_n$ only takes on the values $1$ and $-1$ and let
$p=P(\xi_n=1)$. Now $X_n$ is a non-symmetric random walk. 
Let $a,b$ be integers with $a < 0 < b$. We start the walk at $0$ and
$X_n$ is the position at time $n$. Let $\tau$ be the time when we first hit
$a$ or $b$. Assume that the hypotheses of the optional stopping theorem are
satisfied. Find the probability the walk hits $a$ before it hits $b$, i.e.,
find $P(X_\tau=a)$. 

\bigskip

\no 6. Let $X$ be a random variable with $E[X^2]<\infty$. 
The conditional variance of $X$ given $\field$ is the random variable
defined by 
\bea
Var(X|\field) = E[X^2|\field] - \left(E[X|\field]\right)^2
\eea
Let $X_k$ be an independent sequence of
random variables with $E[X_k^2]<\infty$. Define
$S_n =\sum_{k=1}^n X_k$, and let $\f_n$ be the 
$\sigma$-field generated by $X_1,X_2,\cdots,X_n$. 
Prove that $Var(S_{n+1}|\f_n)=Var(X_{n+1})$ a.s. 

\newpage

\centerline{\bf Some popular RV's }

\medskip

\no $\mu$ is the mean, $\sigma^2$ is the variance, $M(t)=E[e^{tX}]$.

\smallskip

\no {\bf Binomial (2 parameters, $p \in [0,1]$ and positive integer $n$) :} 

$$P(X=k)= {n \choose k} \, p^k \, (1-p)^{n-k}, \quad k=0,1,2,\dots,n$$
$$\mu=np, \quad \sigma^2=np(1-p), \quad M(t) = (pe^t + 1- p)^n$$
\smallskip

\no {\bf Geometric (1 parameter $p \in (0,1]$) :} 
$$P(X=k)=p (1-p)^{k-1}, \quad k=1,2,\dots $$
$$\mu={1 \over p}, \quad \sigma^2 = {1-p \over p^2}, \quad
M(t)= {p e^t \over 1-(1-p)e^t } $$

\smallskip

\no {\bf Poisson (1 parameter $ \lambda >0$) :} 

$$P(X=k)= {e^{-\lambda} \lambda^k \over k!}, \quad k=0,1,2,\dots \qquad \qquad 
\mu= \lambda, \quad \sigma^2=\lambda, \quad M(t)= \exp(\lambda(e^t-1))$$

\no {\bf Exponential (1 parameter, $\lambda>0$) : }

$$f(x)= \lambda e^{-\lambda x}, \quad x \ge 0, \qquad
\mu= {1 \over \lambda}, \quad \sigma^2={1 \over \lambda^2}, \quad
M(t)= {\lambda \over \lambda-t}$$


\no {\bf Normal (2 parameters, $\mu$ and $\sigma^2>0$) : }

$$f(x)= {1 \over \sigma \sqrt{2 \pi}} 
\exp \left( -{(x-\mu)^2 \over 2 \sigma^2} \right), \qquad
M(t)= \exp(\mu t + {1 \over 2} \sigma^2 t^2)$$

\smallskip

\no {\bf Gamma (2 parameters, $\lambda>0$, $w>0$): }

$$f(x)= {\lambda^w \over \Gamma(w)} x^{w-1} e^{-\lambda x},
\quad x \ge 0$$
$$\mu= {w \over \lambda}, \quad \sigma^2={w \over \lambda^2}, \quad
M(t)= \left({\lambda \over \lambda-t}\right)^w $$

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