Assignment 2

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$\newcommand{\v}[1]{\boldsymbol{#1}}$ $\newcommand{\ev}{~\mathrm{e.v.}}$ $\newcommand{\rightarrowD}{\stackrel{d}{\longrightarrow}}$ $\newcommand{\rightarrowP}{\stackrel{P}{\longrightarrow}}$ $\newcommand{\rightarrowA}{\stackrel{a.s.}{\longrightarrow}}$ $\newcommand{\rightarrowL}[1]{\stackrel{L_#1}{\longrightarrow}}$ Each problem carries 1 or 2 marks. Please submit your solutions in handwritten hard copy to your TAs on or before Oct 09, 2026.

EXERCISE 1: $X$ is a nonnegative random variable with finite second moment. Show that $P(X>0)\geq \frac{E^2(X)}{E(X^2)}.$

[1]

EXERCISE 2: We can define a set called $\{A_n\ev\}$ (where $\ev$ is the abbreviation of eventually) as $$\{A_n\ev\} = \cup_n \cap_{k\geq n} A_k.$$ Let $P\{A_n\ev\}=1.$ Show that $P(A_n)\rightarrow 1.$

[1]

EXERCISE 3: Consider $[-1,1]$ as a 1-dimensional dart board, 0 being the bull's eye. A drunkard is throwing darts at this board hitting the board at any random points (all points equally likely). After the each attempt he gets a prize if he hits within $\frac 1n$ distance of the bull's eye (if it is the $n$-th attempt). What is the chance that he gets infinitely many prizes? [Assume infinite life for the guy, but don't drink to it!]

[1]

EXERCISE 4: If $X_n\rightarrowP X,$ show that $|X_n|\rightarrowP |X|.$ Prove it directly using the definition (and not the continuous mapping theorem).

[1]

EXERCISE 5: If $X_n\rightarrowD 5,$ then show that $X_n\rightarrowP 5.$

[1]

EXERCISE 6: (DCT) Let $(X_n),X$ be random variables such that $X_n\rightarrowA X$ and $\forall n\in{\mathbb N}~~|X_n|\leq Y$ for some random variable $Y$ with $E(Y) < \infty.$ Then show that $X_n\rightarrowL 1 X.$

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EXERCISE 7:  Let $(X_n)$ be iid $Poi(\lambda).$ Let $Y_n = \frac 1n\sum_1^n X_k^3.$ Show that $(Y_n)$ converges almost surely. Find the limit.

[1+1]

EXERCISE 8: Let $X_1,X_2,...$ be iid with some distribution having finite moments $E(X_1^k) = \mu_k$ for $k=1,2,3,4.$ We are interested in the sample $CV$ of $X_1,...,X_n:$ $$CV_n = \frac{S_n}{\bar X_n},$$ where $\bar X_n$ and $S^2_n$ are the sample mean and variance of $X_1,...,X_n.$ Use the multivariate CLT and delta method to obtain the asymptotic distribution of $CV_n$.

[2]

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