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

Contents

- 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 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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What is ans of no. 15?

For Q3: the set of points A,B,C,D for D=(-2,1) do not form a parallelogram

Parallelogram is possible only for options A(4,1) and option B(-2,-3) but since A,B,C,D are in clockwise in same order, it should be option A(4,1)

For Q9 distance between the centres of two circles C1(2,2) and C2(-2,-1) id 5=sum of radii(2+3) so the circles touch extrenally, no. of common tangents should be 3

For Q10 the determinant should be 1, can be verified as A^17=A,A^10=A^2: and A^10+A^10-I={(1,4,10),(0,-1,0),(0,0,-1)} hence det=product of disgonal elements =1 Answer should be A(1)

For Q15, each element of A consists of k ones followed by n−k zeroes, where k∈{0,1,…,n}, hence there are n+1 possible elements (option B)

Thanks, Ishan for such a valuable suggestion and being a student of Cheenta. We will make the changes and will discuss the same in our upcoming classes.

Q.15 pls elaborate a little not getting it

I am getting Q1 to be c but here it is b may be I am wrong it will be helpful if you pls tell the logic

The solution for subjective part is not there. From where can I get it?

Even my answer is coming out to be (c) in Q1

what do you expect the cut off to be ?

when can we expect the results to come out ?

No answers ? No expected date ?

I think the 2020 paper was slightly harder than 2019. So , the cutoff should be lesser this year. Of course , all of this is just my opinion. Anything can happen.

What is your opinion on the cutoff ?

I think atleast 4 psb questions will suffice.

Any info about when the results will come out ?

No idea. Are you expecting a good result?

@Anton , what is your opinion on exam difficulty ?

PSB is a bit difficult than previous year papers for sure.

Not sure. Depends on how they grade psb.

How did you do ? I think 60+ would be a good score in psb, assuming you score 90+ in psa.

How did you do ?

Hey Anton , how much do you think will be the cutoff for PSA + (2.5)*PSB ? Also , how many marks are you getting in PSA ?

I am getting 105 in PSA and expecting 70-80 in PSB . Is there any chance for me?

https://artofproblemsolving.com/community/c7h2301933_isi_mstat_2020_

We can discuss here.

This group has been deleted it seems

ISI is extremely late in declaring the result. Don't know what is wrong with them. Did anyone contact them in any way to get some info about results ? Really frustrating...

Any update regarding the result??