27
Dec 22

## Final exam in Advanced Statistics ST2133, 2022

Unlike most UoL exams, here I tried to relate the theory to practical issues.

KBTU International School of Economics

Compiled by Kairat Mynbaev

The total for this exam is 41 points. You have two hours.

Everywhere provide detailed explanations. When answering please clearly indicate question numbers. You don’t need a calculator. As long as the formula you provide is correct, the numerical value does not matter.

Question 1. (12 points)

a) (2 points) At a casino, two players are playing on slot machines. Their payoffs $X,Y$ are standard normal and independent. Find the joint density of the payoffs.

b) (4 points) Two other players watch the first two players and start to argue what will be larger: the sum $U = X + Y$ or the difference $V = X - Y$. Find the joint density. Are variables $U,V$ independent? Find their marginal densities.

c) (2 points) Are $U,V$ normal? Why? What are their means and variances?

d) (2 points) Which probability is larger: $P(U > V)$ or $P\left( {U < V} \right)$?

e) (2 points) In this context interpret the conditional expectation $E\left( {U|V = v} \right)$. How much is it?

Reminder. The density of a normal variable $X \sim N\left( {\mu ,{\sigma ^2}} \right)$ is ${f_X}\left( x \right) = \frac{1}{{\sqrt {2\pi {\sigma ^2}} }}{e^{ - \frac{{{{\left( {x - \mu } \right)}^2}}}{{2{\sigma ^2}}}}}$.

Question 2. (9 points) The distribution of a call duration $X$ of one Kcell [largest mobile operator in KZ] customer is exponential: ${f_X}\left( x \right) = \lambda {e^{ - \lambda x}},\,\,x \ge 0,\,\,{f_X}\left( x \right) = 0,\,\,x < 0.$ The number $N$ of customers making calls simultaneously is distributed as Poisson: $P\left( {N = n} \right) = {e^{ - \mu }}\frac{{{\mu ^n}}}{{n!}},\,\,n = 0,1,2,...$ Thus the total call duration for all customers is ${S_N} = {X_1} + ... + {X_N}$ for $N \ge 1$. We put ${S_0} = 0$. Assume that customers make their decisions about calling independently.

a) (3 points) Find the general formula (when ${X_1},...,{X_n}$ are identically distributed and $X,N$ are independent but not necessarily exponential and Poisson, as above) for the moment generating function of $S_N$ explaining all steps.

b) (3 points) Find the moment generating functions of $X$, $N$ and ${S_N}$ for your particular distributions.

c) (3 points) Find the mean and variance of ${S_N}$. Based on the equations you obtained, can you suggest estimators of parameters $\lambda ,\mu$?

Remark. Direct observations on the exponential and Poisson distributions are not available. We have to infer their parameters by observing ${S_N}$. This explains the importance of the technique used in Question 2.

Question 3. (8 points)

a) (2 points) For a non-negative random variable $X$ prove the Markov inequality $P\left( {X > c} \right) \le \frac{1}{c}EX,\,\,\,c > 0.$

b) (2 points) Prove the Chebyshev inequality $P\left( {|X - EX| > c} \right) \le \frac{1}{c^2}Var\left( X \right)$ for an arbitrary random variable $X$.

c) (4 points) We say that the sequence of random variables $\left\{ X_n \right\}$ converges in probability to a random variable $X$ if $P\left( {|{X_n} - X| > \varepsilon } \right) \to 0$ as $n \to \infty$ for any $\varepsilon > 0$.  Suppose that $E{X_n} = \mu$ for all $n$ and that $Var\left(X_n \right) \to 0$ as $n \to \infty$. Prove that then $\left\{X_n\right\}$ converges in probability to $\mu$.

Remark. Question 3 leads to the simplest example of a law of large numbers: if $\left\{ X_n \right\}$ are i.i.d. with finite variance, then their sample mean converges to their population mean in probability.

Question 4. (8 points)

a) (4 points) Define a distribution function. Give its properties, with intuitive explanations.

b) (4 points) Is a sum of two distribution functions a distribution function? Is a product of two distribution functions a distribution function?

Remark. The answer for part a) is here and the one for part b) is based on it.

Question 5. (4 points) The Rakhat factory prepares prizes for kids for the upcoming New Year event. Each prize contains one type of chocolates and one type of candies. The chocolates and candies are chosen randomly from two production lines, the total number of items is always 10 and all selections are equally likely.

a) (2 points) What proportion of prepared prizes contains three or more chocolates?

b) (2 points) 100 prizes have been sent to an orphanage. What is the probability that 50 of those prizes contain no more than two chocolates?