1. Let $X_{1}, X_{2} \ldots$ in Geometric $(p)$. Let $N \sim$ Poisson $(\lambda)$, with $N$ independent of all of the $X_{i}^{\prime} s$. Let $S_{N}=X_{1}+\ldots+X_{N}$
(a)  Find $P\left(S_{N}=0\right)$.
(b)  Find $\operatorname{Cov}\left(S_{N}, N\right)$.
2. Let $\rho \in(-1,1), \sigma>0$, and $\tau>0 .$ Suppose that $Z \sim N\left(0, \tau^{2}\right)$, and $X_{1}$ is a $N\left(0, \sigma^{2}\right)$ random variable that is independent of $Z$. Let $X_{2}=\rho X_{1}+Z$.
(a)  Find the joint distribution of $X_{1}$ and $X_{2}$.
(b)  State a condition so that $X_{1}$ and $X_{2}$ have the same marginal distributions if and only if that condition is satisfied.
(c)  State a condition so that $X_{1}$ and $X_{2}$ are independent if and only if those conditions are satisfied.
(d)  Show that if $X_{1}$ and $X_{2}$ are i.i.d., then $P\left(X_{1}<X_{2}\right)=0.5$.
3. Let $U \sim$ Uniform $(0,1)$.
(a)  Let $X_{\mathrm{n}}$ be a discrete random variable with $P\left(X_{n}=\frac{i}{n}\right)=\frac{1}{n}$ for $i=1,2, \ldots, n$. Show that $X_{\mathrm{a}} \stackrel{4}{\rightarrow} U$
(b)  Find the asymptotic distribution of $2 \min \left(X_{n}, 1-X_{n}\right)$.
4. Suppose that $X_{1}, X_{2}, \ldots$, are iid with $E\left[X_{i}\right]=\mu$ and $\operatorname{Var}\left[X_{i}\right]=\sigma^{2}<\infty$. Consider the sequence of random variables $Y_{1}, Y_{2}, \ldots$, where
$$Y_{n}=\left{\begin{array}{ll} 14, & \text { if } n=1,2, \ldots, 10^{10} \ \frac{1}{n} \sum_{j=1}^{n} X_{j}, & \text { if } n=10^{10}+1,10^{10}+2, \ldots \end{array}\right.$$
(a)  What does $Y_{n}$ converge in probability to?
(b)  Find the asymptotic distribution of $\sqrt{n}\left(Y_{n}-\mu\right)$.

Let $\theta>0$ be a constant. Let $X_{1}, X_{2} \ldots$ be i.i.d. with pdf $f(x)=\left{\begin{array}{ll}\theta x^{8-1}, & 0<x<1 \text { , } \ 0, & \text { otherwise. }\end{array}\right.$

(a)  Find the distribution of $-\log \left(X_{1}\right)$.

(b)  Find the asymptotic distribution of $\sqrt{n}\left(\frac{1}{n} \sum_{i=1}^{\mathrm{u}} \log \left(X_{i}\right)+\frac{1}{0}\right)$.

(c)  Find the asymptotic distribution of $\sqrt{n}\left(1+\frac{n}{\sum \sum_{i=1}^{n} \log \left(Y_{i}\right)}\right)$.

(d)  Find the asymptotic distribution of $\sqrt{n}\left(\left(\prod_{i=1}^{n} X_{i}\right)^{\frac{1}{n}}-e^{-1}\right)$.