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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.

Contents

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.

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).

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 $\frac{1}{\sqrt{k+1}}$.

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)

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].

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

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 $\beta$.

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 $\alpha$.

- 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 PSA Answer Key

Click on the links to learn about the detailed solution.

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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