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 \(L_{2}\).
**S****olution** - 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

Objective Answer Key

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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