Assignment 1

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$\newcommand{\v}[1]{\boldsymbol{#1}}$ Each problem carries two marks. Please submit your solutions in handwritten hard copy to your TAs on or before Aug 31, 2026.

EXERCISE 1: Let $(X,Y)$ be jointly distributed with density $$f(x,y) = \left\{\begin{array}{ll}x+y &\text{if }x,y\in[0,1]\\ 0&\text{otherwise.}\end{array}\right..$$ Find the density of $(X^2+Y, Y).$ Do not forget to mention the support.

EXERCISE 2:  If $X_1,...,X_{10}$ are iid $N(\mu, \sigma^2)$ where $\mu\in{\mathbb R}$ and $\sigma^2>0$ are both unknown. Find $a > 0$ such that $P\left(-a \leq \frac{\hat \mu-\mu}{\hat \sigma} \leq a\right) = 0.95,$ where $\hat \mu = \bar X$ and $\hat \sigma^2 = \frac{1}{9}\sum_1^{10}(X_i-\bar X)^2.$ Your answer should be in terms of $F_k(\cdot),$ the distribution function of the $t_{(k)}$ distribution.

EXERCISE 3: A branching process has progeny distribution $$P(X=k) = \left\{\begin{array}{ll}\frac 15&\text{if }k=0,1,2,3,4\\ 0&\text{otherwise.}\end{array}\right..$$ Show that the extinction probability is $ < 1.$

EXERCISE 4: If $\v X$ has $N_3(\v \mu, \Sigma)$ distribution, where $\mu= \left[\begin{array}{ccccccccccc}1\\2\\3 \end{array}\right]$ and $$\Sigma = \left[\begin{array}{ccccccccccc}1 & 0 & 0\\0 & 2 & 0\\0 & 0 & 3 \end{array}\right],$$ then prove or disprove the statement:

$X_1+X_2+X_3$ and $X_1-2X_2+X_3$ are independent.

EXERCISE 5: Let $X$ be any nonnegative random variable with finite third moment (i.e., $E(X^3)$ is a finite number). Then show that $$\forall \epsilon>0~~P(X\geq \epsilon) \leq \frac{E(X^3)}{\epsilon^3}.$$

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