Description

G V V Sharma∗

1. There are 30 questions in a certain multiple choice examination paper. Each question has 4 options and exactly one is to be marked by the candidate. Three candidates A,B,C mark each of the 30 questions at random independently. The probability that all the 30 answers of the three students match each other perfectly is?
2. Consider a Markov Chain with state space
{0,1,2,3,4} and transition matrix
0
1
P =
2
3
4 0
 1
100/3
0 1
0
1/3
1/3
0
0 2
0
1/3
1/3
1/3
0 3 0
0
1/3
1/3
0 4
0  / 
0
0
1 31
3. Consider the function f(x) defined as f(x) = ce−x4, x ∈ R. For what value of c is f a probability density function?
a) Γ(214)
b) Γ(414)
c) Γ(313)
d)
4. A random sample of size 7 is drawn from a distribution with p.d.f.
 fθ(x) = 
0, otherwise
and the observations are 12, -54, 26, -2, 24, 17, -39. What is the maximum likelihood estimation of θ.
*The author is with the Department of Electrical Engineering, Indian Institute of Technology, Hyderabad 502285 India e-mail: gadepall@iith.ac.in. All content in this manual is released under GNU GPL. Free and open source.
a) 12
b) 24
c) 26
d) 27
5. Let X1, X2, X3, X4, X5be i.i.d random variables having a continuous distribution function.
Then P(X1 > X2 > X3 > X4 > X5|X1 = max(X1, X2, X3, X4, X5)) equals
a)
b)
c)
d)
6. Suppose (X,Y) follows bivariate normal distribution with means µ1,µ2, standard deviations σ1, σ2 and correlation coefficient ρ, where all parameters are un-known. Then, testing Ho: σ1 = σ2 is equivalent to testing the independence of
a) X and Y
b) X and X-Y
c) X+Y and Y
d) X+Y and X-Y
7. Let X1, X2… be the sequence of independent normally distributed random variables with mean1 and variance 1. Let N be the poisson random variable with mean 2, independent of X1, X2….. Then the variance of X1 + X2 +……..
+XN+1 is a) 3
b) 4
c) 5
d) 9
8. Suppose r1.23 and r1.234 are sample multiple correlation coefficients of X1 on X2, X3 and X1 on X2, X3, X4 respectively. Which of the following is possible?
a) r1.23 = −0.3 and r1.234 = 0.7
b) r1.23 = 0.7 and r1.234 = 0.3
c) r1.23 = 0.3 and r1.234 = 0.7
d) r1.23 = 0.7 and r1.234 = −0.3
9. Suppose X∼ Cauchy(0,1). Then the distribution
of 11+−XX is?
10. There are two sets of observations on a random vector (X,Y). Consider a simple linear regression model with an intercept for regressing Y on X. Let βi be the least square estimate of the regression coefficient obtained from the ith (i=1,2) set consisting of niobservations (n1,n2 > 2). Let β0 the least square estimate obtained from the pooled sample size n1 + n2. If it is known that β1 > β2 > 0 , which of the following statements is true ?
a) β2 < β0 < β1
b) β0may lie outside (β2,β1)but cannot exceed β1 + β2
d) β0 can be negative
11. Let X and Y be i.i.d random variables uniformly distributed on (0,4). Then P(X > Y|X < 2Y) is
12. A sample of size n = 2 is drawn from a population of size N = 4 using probability proportional to size without replacement sampling scheme, where the probabilities proportional to size are
The probability of inclusion of unit 1 in the sample is
i 1 2 3 4
Pi 0.4 0.2 0.2 0.2
TABLE 12
a) 0.4
b) 0.6
c) 0.7
d) 0.75
! X
13. Suppose that has a bivariate density f = Y
1 1
2 f1 + 2 f2, where f1 and f2 are respectively, the densities of bivariate normal distribution
!
P µ2,P), with µ1 = 1 , µ2 =
N(µ1, ), and N(
1
!
−−11 and P = I2, the 2 × 2 identity matrix.
Then which of the following is correct ? a) X and Y are positively correlated
b) X and Y are negatively correlated
c) X and Y are uncorrelated but they are not independent
d) X and Y are independent
14. Suppose Xn is a Markov chain with 3 states and a transition probability matrix
1 1 1
P = 3 3 03 (0.0.1)

  
0 0 1
Then which of the following statement is true: a) Xn is irreducible
b) Xn is recurrent
c) Xn does not admit a stationary probability distribution
d) Xn has an absorbing state
15. Let X1, X2,…., Xn be a random sample from fθ(x), a probability density function or a prob-
ability mass function. Defined
Pn (X −X)
s , Xn
Then s2n is unbiased for θ if: a) f
b) fθ,θ > 0
−x
c) f ,θ > 0
d) fθ(x) = θe−θxx > 0,θ > 0
16. Given the observations
0.8,0.71,0.9,1.2,1.68,1.4,0.88,1.62 from the uniform distribution on (θ − 0.2,θ + 0.8) with −∞ < θ < ∞, which of the following is a maximum likelihood estimate for θ ? a) 0.7
b) 0.9
c) 1.1
d) 1.3
17. Consider a linear model Yl = θ1 +θ2 +l for l = 1,2 and Yl = θ1 −θ3+ for l = 3,4, where l are
independent with R for i = 1, … 4, and θ1,…θ3 ∈ R. Which of the following parametric function is estimable? a) θ1 + θ3
b) θ2 − θ3
c) θ2 + θ3
d) θ1 + θ2 + θ3
18. To test the hypothesis H0 against H1 using the test statistic T, the proposed test procedure is not to support H0 if T is large. Based on a given sample, the p value of the test statistic is computed to be 0.05 assuming that the distribution of T is N(0,1) under H0. If the distribution of T under H0 is the t-distribution with 10 degrees of freedom instead, the p-value will be :
19. Let X ≥ 0 be a random variable on (Ω, F , P) with E(X) = 1. Let A ∈ F be an event with 0 < P(A) < 1. Which of the following defines another probability measure on (Ω, F )?
a) Q(B) = P(A ∩ B) ∀B ∈ F
b) Q(B) = P(A ∪ B) ∀B ∈ F
c) Q(B) = E(XIB) ∀B ∈ F

P(A/B) P(B) > 0
d) Q(B) = 
0 P(B) = 0
20. Consider a Markov Chain having state space S = 1,2,3,4 with a transation probablity matrix P = (pi,j) given by
1 2 3 4

P = 12 12 00



3 31 0 
4  
 0 0
Then
a) , P∞n=0 p(2n,2) = ∞
b) , P∞n=0 p(2n,2) < ∞
c) , P∞n=0 p(2n,2) = ∞
d) , P∞n=0 p(2n,2) < ∞
21. Let X1, X2… be a sequence of independent normally distributed random variables with mean 1 and variance 1. Let N be a Poisson random variable with mean 2 , independent of X1, X2…. Then the variance of X1 + X2 +…+XN+1 is a) 3
b) 4
c) 5
d) 9
22. Let (X1,Y1),(X2,Y2),(X3,Y3)…(Xn,Yn) be n independent observations from a distribution. Let rp be the product moment correlation coefficient and rs be the rank correlation coefficient computed based on n observations. Which of the following statements is correct. a) rp > 0 implies rs > 0
b) rs > 0 implies rp > 0
c) rp = 1 implies rs = 1
d) rs = 1 implies rp = 1
23. Let X and Y be i.i.d uniform (0,1) random variables. Let Z = max(X,Y) and W = min(X,Y). Then P((Z-W)> 0.5) =
24. In a 24 experiment with two blocks and factors A, B, C and D, one block contains the following treatment combinations a,b,c,ad,bd,cd,abc,abcd. Which of the following effects is confounded?
a) ABC
b) ABD
c) BCD
d) ABCD
25. Suppose that the lifetime of an electric bulb follows an exponential distribution with mean θ hours. In order to estimate θ, n bulbs are switched on at the same time. After t hours, n− m(> 0) bulbs were found to be in functioning state. If the lifetimes of the other m(> 0) bulbs are noted as x1, x2, x3,…, xm, respectively, then the maximum likelihood estimate of θ is given by –
26. If X ∼ Np(0, I) and Apxp is an idempotent matrix with rank(A) = k < p, then which of the following statements is correct?
a) XX00AXX ∼ kpFk,p
b) XX00AXX ∼ p−kk Fk,p−k
c) XX00AXX ∼ Beta2k, 2p
d) XX00AXX ∼ Beta2k, p−2k
27. A standard fair die is rolled until some face other than 5 or 6 turns up. Let X denote the face value of the last roll, and A = [X is even] and B = [X is at most 2]. Then a) P(AT B) = 0
b) P(AT B) = 1/6
c) P(AT B) = 1/4
d) P(AT B) = 1/3
28. A simple random sample of size n will be drawn from a class of 125 students, and the mean mathematics score of the sample will be computed. If the standard error of the sample mean for ”with sampling” is twice as much as the standard error of the sample mean for ”without replacement sampling”, the value of n is-
a) 32
b) 63
c) 79
d) 94
29. Twenty items are put in a life testing experiment starting at time 0. The failure times of the items are recorded in a sequential manner. The experiment stops if all the twenty items fails or a pre-fixed time T> 0 is reached, whichever is earlier. If the lifetimes of the items are independent and identically distributed exponential random variables with mean θ, where 0 < θ < 10, then which of the following statements are correct ?
a) The MLE of θ always exists.
b) The MLE of θ does not exist.
c) The MLE of θ is an unbiased estimator of θ if it exists.
d) The MLE of θ is bounded with probability
1, if it exists.
30. Suppose that the lifetime of an electric bulb follows an exponential distribution with mean θ hours. In order to estimate θ, n bulbs are switched on at the same time. After t hours, n-m(>0) bulbs are found in functioning state. If the lifetimes of other m(>0) bulbs are noted as x1, x2,…., xm respectively, then the maximum likelihood estimate of θ is given by –
31. In an airport, domestic passengers and international passengers arrive independently according to Poisson processes with rates 100 and 70 per hour. If it is given that the total no. of passengers (domestic and international) arriving in that airport between 9:00 AM and 11:00 AM on a particular day was 520, then what is the conditional distribution of the no. of domestic passengers arriving in this period ?
a) Poisson (200)
b) Poisson (100)
c) Binomial (520, )
d) Binomial (520, )
32. The covariance matrix of four dimensional random vector X is of the form
 
1
ρρρ ρ
1 ρ ρ ρ ρ
1 ρ ρ , Where ρ < 0
ρ ρ 1
If v is the variance of the first principle component then,
a) v cannot exceed
b) v can exceed but cannot exceed
c) v can exceed but cannot exceed
d) v can exceed
33. Consider the problem of estimation of a parameter θ on the basis of X, where X∼N(θ,1) and −∞ < θ < ∞. Under squared error loss, X has uniformly smaller risk than that of kX, for
a) k < 0
b) 0 < k < 1
c) k > 1
d) no value of k
34. Suppose A, B,C are events in a common probability space with P(A) = 0.2, P(B) =
0.2, P(C) = 0.3, P(AB) = 0.1, P(AC) = 0.1, P(BC) = 0.1 Which of the following are possible values of P(A + B + C)? a) 0.5
b) 0.3
c) 0.4
d) 0.9
35. Let X1, X2, X3..Xn be i.i.d. uniform (θ1,θ2) random variables, where θ1 < θ2 are unknown parameters. Which of the following is an ancillary statistic?
X(k)
a) X(n) for any k < n
X(n)−X(k)
b) X(n) for any k < n
X(k)
c) X(n)−X(k) for any k < n
X(k)−X(1)
d) X(n)−X(k) for any k < n where 1 < k < n
37. Suppose X1,…, Xn are i.i.d. random vectors from Np(0,Σ). Let l ∈ Rp, E(Σni=1ltXiXitl) = c and E XiXit) = A.
Which of the following statements are necessarily true? a) c = ltl
b) lt(Σni=1XiXit)l follows a chi-squared distribution
c) lt(Σni=11XiXit)l and lt XiXit)l are indepen-
dently distributed for 1 ≤ n1 ≤ n − 1
d) A =Σ
38. Arrival of customers in a shop is a Poisson process with intensity λ = 2. Let X be the number of customers entering during the time interval (1, 2) and let Y be the number of customers entering during the time interval (5,10). Which of the following are true?
a) P(X = 0|(X + Y = 12)) = (56)12
b) X and Y are in-dependant
c) X + Y is a Poisson with parameter 6
d) X − Y is a Poisson with parameter 8
39. Let the joint distribution of (X,Y) be bivariate 0!
normal with the mean vector and variance-
0
1 ρ!
covariance matrix , where -1 < ρ < 1. Let ρ 1
φρ(0,0) = P(X ≤ 0,Y ≤ 0). Then the Kendall’s τ coefficient between X and Y equals a) 4φρ(0,0) − 1
b) 4φρ(0,0)
c) 4φρ(0,0) + 1
d) φρ(0,0)
40. Let the joint distribution of (X,Y) be bivariate 0!
normal with the mean vector and variance-
0
1 ρ!
covariance matrix , where -1 < ρ < 1. Let ρ 1
φρ(0,0) = P(X ≤ 0,Y ≤ 0). Then the Kendall’s τ coefficient between X and Y equals a) 4φρ(0,0) − 1
b) 4φρ(0,0)
c) 4φρ(0,0) + 1
d) φρ(0,0)
41. Consider a markov chain with 5 states 1,2,3,4,5 and transition matrix P =
1/2 0 0 1/2 0 
10/5 11//75 10/5 10/5 16//57 (0.0.2)

10/3 50/8 00 20/3 30/8
Which of the following is true?
a) 3 and 1 communicating class
b) 1 and 4 communicating class
c) 4 and 2 communicating class
d) 2 and 5 communicating class