## ISI MStat Entrance 2020 Problems and Solutions

This post contains Indian Statistical Institute, ISI MStat Entrance 2020 Problems and Solutions. Try to solve them out.

## Subjective Paper – ISI MStat Entrance 2020 Problems and Solutions

• Let $f(x)=x^{2}-2 x+2$. Let $L_{1}$ and $L_{2}$ be the tangents to its graph at $x=0$ and $x=2$ respectively. Find the area of the region enclosed by the graph of $f$ and the two lines $L_{1}$ and $L_{2}$.

Solution
• Find the number of $3 \times 3$ matrices $A$ such that the entries of $A$ belong to the set $\mathbb{Z}$ of all integers, and such that the trace of $A^{t} A$ is 6 . $\left(A^{t}\right.$ denotes the transpose of the matrix $\left.A\right)$.

Solution
• Consider $n$ independent and identically distributed positive random variables $X_{1}, X_{2}, \ldots, X_{n},$ Suppose $S$ is a fixed subset of ${1,2, \ldots, n}$ consisting of $k$ distinct elements where $1 \leq k<n$
(a) Compute $\mathbb{E}\left[\frac{\sum_{i \in S} X_{i}}{\sum_{i=1}^{n} X_{i}}\right]$

(b) Assume that $X_{i}$ ‘s have mean $\mu$ and variance $\sigma^{2}, 0<\sigma^{2}<\infty$. If $j \notin S,$ show that the correlation between $\left(\sum_{i \in S} X_{i}\right) X_{j}$ and $\sum_{i \in S} X_{i}$ lies between -$\frac{1}{\sqrt{k+1}} \text { and } \frac{1}{\sqrt{k+1}}$.

Solution
• Let $X_{1,} X_{2}, \ldots, X_{n}$ be independent and identically distributed random variables. Let $S_{n}=X_{1}+\cdots+X_{n}$. For each of the following statements, determine whether they are true or false. Give reasons in each case.

(a) If $S_{n} \sim E_{x p}$ with mean $n,$ then each $X_{i} \sim E x p$ with mean 1 .

(b) If $S_{n} \sim B i n(n k, p),$ then each $X_{i} \sim B i n(k, p)$

Solution
• Let $U_{1}, U_{2}, \ldots, U_{n}$ be independent and identically distributed random variables each having a uniform distribution on (0,1) . Let $X=\min \{U_{1}, U_{2}, \ldots, U_{n}\}$, $Y=\max \{U_{1}, U_{2}, \ldots, U_{n}\}$

Evaluate $\mathbb{E}[X \mid Y=y]$ and $\mathbb{E}[Y \mid X=x]$.

Solution
• Suppose individuals are classified into three categories $C_{1}, C_{2}$ and $C_{3}$ Let $p^{2},(1-p)^{2}$ and $2 p(1-p)$ be the respective population proportions, where $p \in(0,1)$. A random sample of $N$ individuals is selected from the population and the category of each selected individual recorded.

For $i=1,2,3,$ let $X_{i}$ denote the number of individuals in the sample belonging to category $C_{i} .$ Define $U=X_{1}+\frac{X_{3}}{2}$

(a) Is $U$ sufficient for $p ?$ Justify your answer.

(b) Show that the mean squared error of $\frac{U}{N}$ is $\frac{p(1-p)}{2 N}$

Solution
• Consider the following model: $y_{i}=\beta x_{i}+\varepsilon_{i} x_{i}, \quad i=1,2, \ldots, n$, where $y_{i}, i=1,2, \ldots, n$ are observed; $x_{i}, i=1,2, \ldots, n$ are known positive constants and $\beta$ is an unknown parameter. The errors $\varepsilon_{1}, \varepsilon_{2}, \ldots, \varepsilon_{n}$ are independent and identically distributed random variables having the probability density function $f(u)=\frac{1}{2 \lambda} \exp \left(-\frac{|u|}{\lambda}\right), \quad-\infty<u<\infty$ and $\lambda$ is an unknown parameter.

(a) Find the least squares estimator of $\beta$.

(b) Find the maximum likelihood estimator of $\beta$.

Solution
• Assume that $X_{1}, \ldots, X_{n}$ is a random sample from $N(\mu, 1),$ with $\mu \in \mathbb{R}$. We want to test $H_{0}: \mu=0$ against $H_{1}: \mu=1$. For a fixed integer $m \in{1, \ldots, n},$ the following statistics are defined:

\begin{aligned}
T_{1} &= \frac{\left(X_{1}+\ldots+X_{m}\right)}{m} \\
T_{2} &= \frac{\left(X_{2}+\ldots+X_{m+1}\right)}{m} \\
\vdots &=\vdots \\
T_{n-m+1} &= \frac{\left(X_{n-m+1}+\ldots+X_{n}\right)}{m}
\end{aligned}

$\operatorname{Fix} \alpha \in(0,1) .$ Consider the test

Reject $H_{0}$ if $\max \{T_{i}: 1 \leq i \leq n-m+1\}>c_{m, \alpha}$

Find a choice of $c_{m, \alpha} \in \mathbb{R}$ in terms of the standard normal distribution function $\Phi$ that ensures that the size of the test is at most $\alpha$.

Solution
• A finite population has $N$ units, with $x_{i}$ being the value associated with the $i$ th unit, $i=1,2, \ldots, N$. Let $\bar{x}{N}$ be the population mean. A statistician carries out the following experiment.

Step 1: Draw an SRSWOR of size $n({1}$ and denote the sample mean by $\bar{X}{n}$

Step 2: Draw a SRSWR of size $m$ from $S_{1}$. The $x$ -values of the sampled units are denoted by $\{Y_{1}, \ldots, Y_{m}\}$

An estimator of the population mean is defined as,

$\widehat{T}{m}=\frac{1}{m} \sum{i=1}^{m} Y_{i}$

(a) Show that $\widehat{T}{m}$ is an unbiased estimator of the population mean.

(b) Which of the following has lower variance: $\widehat{T}{m}$ or $\bar{X}_{n} ?$

Solution

## Objective Paper

 1. C 2. D 3. A 4. B 5. A 6. B 7. C 8. A 9. C 10. A 11. C 12. D 13. C 14. B 15. B 16. C 17. D 18. B 19. B 20. C 21. C 22. D 23. A 24. B 25. D 26. B 27. D 28. D 29. B 30. C

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## How to roll a Dice by tossing a Coin ? Cheenta Statistics Department

How can you roll a dice by tossing a coin? Can you use your probability knowledge? Use your conditioning skills.

Suppose, you have gone to a picnic with your friends. You have planned to play the physical version of the Snake and Ladder game. You found out that you have lost your dice.

The shit just became real!

Now, you have an unbiased coin in your wallet / purse. You know Probability.

### Aapna Time Aayega

starts playing in the background. :p

## Can you simulate the dice from the coin?

Ofcourse, you know chances better than others. :3

Take a coin.

Toss it 3 times. Record the outcomes.

HHH = Number 1

HHT = Number 2

HTH = Number 3

HTT = Number 4

THH = Number 5

THT = Number 6

TTH = Reject it, don’t ccount the toss and toss again

TTT = Reject it, don’t ccount the toss and toss again

Voila done!

What is the probability of HHH in this experiment?

Let X be the outcome in the restricted experiment as shown.

How is this experiment is different from the actual experiment?

This experiment is conditioning on the event A = {HHH, HHT, HTH, HTT, THH, THT}.

$P( X = HHH) = P (X = HHH | X \in A ) = \frac{P (X = HHH)}{P (X \in A)} = \frac{1}{6}$

Beautiful right?

Can you generalize this idea?

## Food for thought

• Give an algorithm to simulate any conditional probability.
• Give an algorithm to simulate any event with probability $\frac{m}{2^k}$, where $m \leq 2^k$.
• Give an algorithm to simulate any event with probability $\frac{m}{2^k}$, where $n \leq 2^k$.
• Give an algorithm to simulate any event with probability $\frac{m}{n}$, where $m \leq n \leq 2^k$ using conditional probability.