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ISI MStat Entrance 2020 Problems and Solutions PSA & PSB

This post contains ISI MStat Entrance PSA and PSB 2020 Problems and Solutions that can be very helpful and resourceful for your ISI MStat Preparation.

ISI MStat Entrance 2020 Problems and Solutions - Subjective Paper


ISI MStat 2020 Problem 1

Let f(x)=x2−2x+2. Let L1 and L2 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 L1 and L2.

Solution

ISI MStat 2020 Problem 2

Find the number of 3×3 matrices A such that the entries of A belong to the set Z of all integers, and such that the trace of AtA is 6 . (At denotes the transpose of the matrix A).

Solution

ISI MStat 2020 Problem 3

Consider $n$ independent and identically distributed positive random variables $X_{1}, X_{2}, \ldots, X_{n}$. Suppose $S$ is a fixed subect of ${1,2, \ldots, n}$ consisting of $k$ distinct ekements where $1 \leq k<n$.
(a) Compute
$$
\mathrm{E}\left[\frac{\sum_{i \in s} X_{i}}{\sum_{i=1}^{\infty} X_{i}}\right]
$$
(b) Assume that $X_{i}$ is 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}X_{i} $ lies between $-\frac{1}{\sqrt{k+1}}$ and $\frac{1}{\sqrt{k+1}}$.

Solution

ISI MStat 2020 Problem 4

Let X1,X2,…,Xn be independent and identically distributed random variables. Let Sn=X1+⋯+Xn. For each of the following statements, determine whether they are true or false. Give reasons in each case.

(a) If Sn∼Exp with mean n, then each Xi∼Exp with mean 1 .

(b) If Sn∼Bin(nk,p), then each Xi∼Bin(k,p)

Solution

ISI MStat 2020 Problem 5

Let U1,U2,…,Un be independent and identically distributed random variables each having a uniform distribution on (0,1) . Let X=min{U1,U2,…,Un}, Y=max{U1,U2,…,Un}

Evaluate E[X∣Y=y] and E[Y∣X=x].

Solution

ISI MStat 2020 Problem 6

Suppose individuals are classified into three categories C1,C2 and C3 Let p2,(1−p)2 and 2p(1−p) be the respective population proportions, where p∈(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 Xi denote the number of individuals in the sample belonging to category Ci. Define U=X1+X32

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

(b) Show that the mean squared error of UN is p(1−p)2N

Solution

ISI MStat 2020 Problem 7

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
$$
f(u)=\frac{1}{2 \lambda} \exp \left(-\frac{|u|}{\lambda}\right),-\infty<u<\infty
$$
and $\lambda$ is an unknown parameter.
(a) Find the least squares estimator of $\beta$.
(b) Find the maximum likelihood estimator of $\beta$.

Solution

ISI MStat 2020 Problem 8

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

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

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

Solution

ISI MStat 2020 Problem 9

  • A finite population has N units, with xi being the value associated with the i th unit, i=1,2,…,N. Let 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 X¯n

    Step 2: Draw an SRSWR of size m from S1. The x -values of the sampled units are denoted by {Y1,…,Ym}

    An estimator of the population mean is defined as,

    Tˆm=1m∑i=1mYi

    (a) Show that Tˆm is an unbiased estimator of the population mean.

    (b) Which of the following has lower variance: Tˆm or X¯n?

    Solution

ISI MStat 2020 - Objective Paper


ISI MStat 2020 PSA Answer Key

Click on the links to learn about the detailed solution.

1. C2. D3. A4. B5. A
6. B7. C8. A9. C10. A
11. C12. D13. C14. B15. B
16. C17. D18. B19. B20. C
21. C22. D23. A24. B25. D
26. B27. D28. D29. B30. C

Please suggest changes in the comment section.

ISI MStat 2020 Probability Problems Discussion [Recorded Class]

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