MA2510-2425A

Final exam (December 10th, 2024, 14:00-17:00)

(30 points) In this exercise, we consider two decks of cards: a "black" deck of 52 cards (containing the values $2, \dots, 10, J, Q, K$ , and $A$ , for each of the four suits $^{1}$ ), and a "red" deck of 32 cards (containing only $7, \dots, 10, J, Q, K$ , and $A$ , of the four suits).
(i) (6 points) We perform the following procedure: we toss a fair coin to select one of the two decks, and we then pick two cards simultaneously (i.e., at the same time), uniformly at random, from the selected deck. Compute the probability that the two cards are of the same value.
(ii) (4 points) We observe that the two cards are of the same value: compute the probability that the black deck was selected.
(iii) (4 points) We now merge and shuffle the two decks, to produce a deck of $52 + 32 = 84$ cards. We then pick two cards simultaneously, uniformly at random, from that large deck. Compute the probability that the two cards are of the same value.
(iv) ( 4 points) We successively pick five cards from the 84 -card deck, with replacement: each of them is chosen uniformly at random, and then put back into the deck, before picking the next card. Let $X_{B}$ be the number of black cards that are picked, and $X_{R}$ be the number of red cards. Determine the distribution of $X_{B}$ , and the distribution of $X_{R}$ .
(v) ( 6 points) Let $X = X_{B} - X_{R}$ . Compute $E [X]$ and $Var (X)$ .
(vi) ( 6 points) We now pick successively $n = 7000$ cards (still with replacement), and we count the number $X_{J Q K}$ of cards which are either $J, Q$ or $K$ (i.e., whose value belongs to ${J, Q, K}$ ). Give an estimate of $P (X_{J Q K} \geq 2100)$ using the Central Limit Theorem, in terms of the cumulative distribution function $Φ$ of a random variable with the standard normal distribution.
(30 points) We consider two jointly continuous random variables $X$ and $Y$ , with joint density function

f_{(X, Y)} (x, y) = c \cdot 1_{| x | + | y | < 1}

for some constant $c \in R$ .
(i) (4 points) Determine the value of $c$ .
(ii) (4 points) Find the marginal probability density function of $X$ .
(iii) (4 points) Compute $E [X]$ .
(iv) ( 6 points) Compute $Var (X)$ .
(v) ( 6 points) Compute $Cov (X, Y)$ . Are the random variables $X$ and $Y$ independent?
(vi) ( 6 points) Compute $E [X^{2} Y^{2}]$ .

^[1]3. (30 points) Let $n \geq 1$ be an integer, and $I_{n} = {1, \dots, n}$ (the set of integers between 1 and $n$ ). We let $P_{n} = P (I_{n})$ be the power set of $I_{n}$ , containing all subsets of $I_{n}$ . In this exercise, we perform the following random experiment. We pick a subset $A$ of $I_{n}$ uniformly at random, that is, all elements of $P_{n}$ are equally likely. We then consider the random variables $X = | A |$ and $Y = | I_{n} ∖ A |$ .
(i) (6 points) Compute $P (X = 0)$ , and then $P (X = 1)$ .
(ii) (6 points) In this question only, we consider $n = 4$ . Draw the cumulative distribution function of $X$ .
(iii) (8 points) In the remainder of the exercise, we now consider an arbitrary integer $n \geq 1$ . Compute $E [X]$ and $Var (X)$ .
(iv) (4 points) Are the two events ${X = 0}$ and ${Y = 0}$ independent?
(v) (2 points) Does $Y$ have the same distribution as $X$ ? Explain.
(vi) (4 points) Compute $Cov (X, Y)$ .
4. (30 points) Let $θ \in R$ be given. We consider a random variable $X$ with probability density function

f_{X} (x) = \frac{3}{(x - θ + 1)^{4}} \cdot 1_{[θ, + \infty)} (x) .

(i) ( 6 points) Compute $E [X]$ .
(ii) ( 8 points) Compute $Var (X)$ .
(iii) (6 points) We now consider a sample $(X_{1}, \dots, X_{n})$ of size $n \geq 1$ : the random variables ${(X_{i})}_{1 \leq i \leq n}$ are independent, and each is continuous with the same distribution as $X$ , i.e. with probability density function $f_{X}$ . Find an estimator of $θ$ by the method of moments.
(iv) ( 4 points) Compute the bias of the estimator found in the previous question.
(v) ( 6 points) Determine the maximum-likelihood estimator ${\hat{θ}}_{MLE}$ of $θ$ . Is it biased?

$^{1}$ clubs ( $%$ ), diamonds ( $⋄$ ), hearts ( $($ ), and spades ( $↑$ ) ↩︎