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This post contains ISI MStat Entrance PSA and PSB 2021 Problems and Solutions that can be very helpful and resourceful for your ISI MStat Preparation.

Contents

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- 49 (Rolle's Theorem)

2. 2 (4 - number of linear constraints)

3. k = 2 (a = -d, and form a biquadratic which has two real solutions)

4. 0 (divide by $x^4$, use $\frac{sinx}{x}$ limit result)

5. $\frac{p}{q}$ must be a rational number. (The product must be a rational number.)

6. $\alpha = 1, \beta =1$ (Use sandwich theorem on an easy inequality on ceiling of x)

7. $\frac{2n}{n+1}$ (Use geometry and definite integration)

8. $2+ \sqrt{5}$ (Just write down the pythagoras theorem in terms of the variables and solve)

9. 10 (Use the roots of unity)

10. $\frac{3}{8}$ (Find out the cases when it is non zero, and use classical probability)

11. $\frac{(n+1)^n}{n!}$ (Use ${{n} \choose {r}}={{n-1} \choose {r-1}}+{{n-1} \choose r}$)

12. $P(\pi)$ is even for all $\pi$. (Observe that there is one more odd than number of evens, so there will be one odd-odd match)

13. is equal to 12. (The $i,j$th element is $a_{ii}b{ij}c{jj}$. Use gp series then.)

14. 200 (Use the fact any permutation can be written as compositions of transpositions. Observe that the given condition is equivalent to that 2 transpositions are not possible)

15. $m_t < \infty$ for all $t \geq 0$ (All monotone functions are bounded on [a,b])

16.$H(x) = \frac{1-F(-x)+ F(x)}{2}$ (If $F(x)$ is right continuous, $F(-x)$ is left continuous.).

17. $\frac{1}{25}$ (Use the distribution function of $\frac{X}{Y}$)

18. 3 (Find the distribution of order statistic, and find the expectation)

19. (II) but not (I) (If $F(x)$ is right continuous, $F(-x)$ is left continuous.).

20. $20\lambda^4$ (Use gamma integral to find the $E(X_{1}^4)$.)

21. The two new observations are 15 and 5. (Use the condition to find two linear equations to find the observations).

22. It is less than 2. (Use the beta coefficients in terms of sample covariance and sample variance, and compare)

23. 4:3 (Use Bayes' Theorem)

24. The two-sample t-test statistic and the ANOVA statistics yield the same power for any non-zero value of $\mu_1 - \mu_2$ and for any $n,m$. (Both the test statistic are one to one function of one another)

25. t³-1 - 2(t-1)

26. $\frac{2 \sum_{i=1}^{n} X_i}{n(n+1)}$ (Use the invariance property of MLE)

27. $Y_1^2 + Y_2^2 + Y_1Y_2$ (Write the bivariate normal distribution in terms of $Y_1, Y_2$ and use Neyman Factorization Theorem.)

28. can be negative (Simson's Paradox)

29. $2z$ (There are three random variables, $N$ = stopping time to get $Y=1$, $Y$ and $X$. Use the conditioning properly. Take your time)

30. $\frac{40}{3}$ (Use the property that Poisson | Poisson in the given problem follows Binomial)

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**ISI MStat PSB 2021** Problem 9**Solution**

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