Name of the University : Indian Statistical Institute
Exam : ISI Admission Test
Document Type : Sample/Previous Year Question Paper
Name of the Subject : JRF Statistics
Year : 2016
Website : http://www.isical.ac.in/~admission/IsiAdmission2017/PreviousQuestion/Questions-Jrf-Stat.html
Download Sample/Previous Years’ Questions :
STA 2016 : https://www.pdfquestion.in/uploads/11352-JRF-Stat2016.pdf
STB 2016 : https://www.pdfquestion.in/uploads/11352-JRF-Stat-2016.pdf
STA 2015 : https://www.pdfquestion.in/uploads/11352-JRF-Stat-2015.pdf
JRF Statistics ISI Admission Sample Questions Paper :
No. of Questions : 8
Time: 2 hours
Instruction :
** Answer as many questions as you can. All questions carry equal weight.
** Do not feel discouraged if you are not able to answer all the questions.
Related : Indian Statistical Institute JRF Entrance Exam Question Paper Model : www.pdfquestion.in/7618.html
** Partial credit may be given for partial answer.
** Full credit will be given for complete and rigorous arguments.
** Write your Name, Registration Number, Test Code, Booklet No. etc., in the appropriate places on the answer-booklet.
** All Rough Work Must Be Done On This Booklet And/Or On The Answer-Booklet.
** You Are Not Allowed To Use Calculators.
1. Consider a random arrangement of 20 boys and 16 girls in a line. Let X be the number of boys with girls on both sides, and Y be the number of girls with boys on both sides. Find E(X + Y).
2. Let X1, …, Xn be independently and identically distributed random variables with distribution function F, where n is an odd integer. Let Y1, …, Y, be independently and identically distributed observations from the empirical distribution function F, associated with X1, …, Xn. Obtain the distribution of the median of Yi, …, Yn.
3. Let X be a non negative random variable such that *(xx)– (*)
(a) Show that X cannot be uniformly distributed over (0,1).
(b) Show that P(X > 1) = 0.
(c) Give an example of a continuous random variable X satisfying the relation in (*).
4. A shopkeeper places an order of one extra item only when his stock of that item at the end of a week is one or less. Let Xn, n >1, denote the number of items in hisstock at the end of the nth week before he decides to place an order or not, with X0 = 0.Assume that the weekly demand for the sale of this item from this shop follows a Poisson distribution with mean X, and the item is sold on demand till the stock lasts.Prove that Xn, n > 1, defines a Markov chain, and then specify its state space and the transition probability matrix. Also derive the stationary probabilities.
5. Let (X1, Y1), …, (X, Y,) be independently and identically distributed observations from the uniform distribution over the triangle ABC, where A = (0,0), B = (a,0), C = (a,b), and a, b are unknown positive constants. Define (U,V,) = (X, Y, ?ix.). Show that, for each i = 1,…, n, the random variable U; is independent of V. Hence or otherwise, find the maximum likelihood estimator of the area of the triangle ABC. . Let Y = e^* + Z, where U, X, Z are independent. Here X is uniformly distributed over (0,1), Z has the standard normal distribution and U has density f(u) = 2w, for 0 < u < 1.
(a) Find the best predictor of Y given X = # when U and Z are unknown.
(b) Find the best linear predictor of Y given X = } when U and Z are unknown.
7. Suppose that you have a random observation X with density f over the real line. Based on X, we want to test 1 – 2/2 1 Ho ; f(a) = Vº /* versus H. : f(t) = T(1 + x2)
(a) Prove that a test which rejects Ho if and only if |X| > dT”(0.975) is a most powerful test of level o = 0.05 for this problem. Here d(a) is the standard normal distribution function.
(b) Find the power of the test described in part (a) above.
8. Consider a completely randomised design (CRD) with t treatments, t > 2, and n = 4t + 3 observations. If we want to minimise the average variance of the best linear unbiased estimators BLUEs) of all the pairwise comparisons of treatment effects, how many observations should be allocated to each treatment?