Shivaji University, Kolhapur

Question Bank For May 2022 (Summer) Examination

B.Sc. (Part-II) (Semester- III) Examination
Subject Code: 73305
Subject Name: STATISTICS (Paper – V) (CBCS-DSC-7C)
Name of the Paper: Probability distributions – I

Multiple Choice Questions (1 Mark each)

Q. Answer the following questions choosing the most correct alternative given below
them.

1) If X and Y are two independent Poisson random variables with parameters 1 and 1
respectively then distribution of X+Y is…
A) Poisson with parameter 2 B) Poisson with parameter 3
C) Poisson with parameter 1 D) none of these
2) If X ~ Poisson (5) then ratio of mean to the variance is?
A) 5 B) 1
C) 100 D) 25
3) If X ~ P(  ) then p.g.f. of X is…

− (1− S )  − (1+ S ) 
A) e B) e

C) e (1− S )  D) e −
4) If X ~ Poisson distribution with parameter 1 then P(X=0) is…
A) e B) 1/e
C) 1 D) none of these
5) If X is a Poisson variate with P[ X =1] = P[X=2] then mean of X is…
A) 1 B) 4
C) 3 D) 2
6) The Poisson distribution is limiting case of binomial distribution when p→ 0 and …
A) n → 0 B) n → 
C) n → p D) n → 1/2

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7) Which of the following distribution has lack of memory property?
A) Poisson distribution B) Geometric distribution
C) Binomial distribution D) none of these
8) If X ~ Geometric distribution with parameter p and P(X > 8 / X >3 )=0.7 then
P(X > 5) is…
A) 0.7 B) 0.3
C) 0.1 D) 0
9) If X ~ G(p) then mean of geometric distribution is…
A) p / q B) q
C) p D) q /p
10) If X ~G(p) and Y ~G(p) are independent variables then X + Y ~…
A) G(p) B) G(q)
C) NBD(2, p) D) NBD(4, q)
11) If X ~ NBD(k, p) then mean of X is …
A) kp B) pq
C) kp/q D) kq/p
12) If X ~ NBD(k, p) it reduces to geometric distribution if …
A) k =1 B) k =0
C) p =1 D) p = 0
13) If X is number of failures before kth success then X follows…….distribution.
A) Poisson B) Geometric
C) Negative Binomial D) None of these
−1
14) The mgf of a random variable X is M X (t ) = (1 − 2t ) . Then E(X) is …
A) 2 B) 5
C) 4 D) ½
15) The cumulant generating function (c.g.f.) of a continuous r. v. X is –log(1–2t) then
E(X) is …
A) 2 B) 5
C) 4 D) 0.5
16) The probability distribution of continuous r. v. X has measures of kurtosis (2) is 3
and fourth central moment(4) is 12 then variance is...
A) 4 B) 2
C) 6 D) 8

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17) If X is a continuous r.v. with p.d.f. f(x) then P(X= 5) is…
A) ½ B) 0
C) infinity D) 1
18) Which of the following is true?
A) First central moment = 0
B) Second Central moment = Second cumulant
C) Third Central moment = Third cumulant
D) All of the these
19) If p.d.f of continuous random variable X is f(x) = 1/2 , if 3 < x < 5 and F(x) be the
cdf of X then F(6) is…..
A) 1 B) –1
C) 0 D) infinity
20) If Q1 and Q3 are first and third quartile of a continuous r. v. X then
P(X < Q1) + P(X > Q3) = …
A) 0.5 B) 0.25
C) 0.125 D) 1
∞
21) If X is a continuous r.v. with p.d.f. f(x) then ∫−∞ 𝑓(𝑥)𝑑𝑥 = ………
A) ½ B) 0
C)  D) 1
22) If X is a continuous r. v. with probability density function(p.d.f.)
𝑘𝑒 −𝑥 0 𝑥 <
𝑓(𝑥) = { then value of k is…
0 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
A) 1 B) 2
C) 3 D) ½
23) If F(x) be the distribution function of continuous random variable X then
P(4  x  10) is equal to:
A) F(10) – F(4) B) F(10) + F(4)
C) F(4) – F(10) D) F(10)* F(4)
24) If X is a continuous r. v. with probability density function(p.d.f.)
kx 2 0  x 1
f ( x) =  Then value of k is --------
 0 otherwise
A) 2 B) 1
C) 3 D) ½

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25) A continuous r.v. X has pdf
2 x 0  x  1
f ( x) = 
 0 otherwise

Then mean of the r.v. X is…

A) 1/2 B) 2/3
C) 1 D) none of these
26) If X is continuous random variable with pdf
2x, 0<x <1
f(x) = { then median of X is…
o, otherwise
1 1
A) B)
2 √2
1
C) D) √2
4

27) A continuous r.v. X has pdf

6 x(1 − x) 0  x  1
f ( x) = 
 0 otherwise

Then mode of the r.v. X is…

A) 1/2 B) 6
C) 0 D) ¼
28) If p.d.f of continuous random variable X is f(x) = 0.25, if 0 < x < 4 then E(X) is…
A) 0.5 B) 2
C) 0 D) 4
29) A continuous r.v. X has mean zero. The expression E (X2) is…
A) µ3 B) Var(X)
C) E(X) D) None of these
30) If X and Y are two independent continuous r. v.’s with mean of X is 5 and mean of Y
is 2 then E(XY) is…
A) 0 B) 7
C) 3 D) 10
31) For the following joint p.d.f. of bivariate continuous r.v. (X, Y) the value of k is…
4𝑘𝑥𝑦 0 < 𝑥 < 1, 0 < 𝑦 < 1
𝑓(𝑥) = {
0 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒

A) 4 B) 1
C) 2 D) 1/3

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32) The joint cumulative distribution function of X and Y i.e. F(x,y) =…
A) P(X = x, Y= y) B) P(X  x, Y  y)
C) P(X  x, Y  y) D) P(X  x, Y  y)
33) If V(X) = V(Y) = Cov(X, Y) then r(X, Y) is -------
A) 1 B) V(X)
C) – 1 D) 1/(V(X)
34) If F(x, y) be the joint cumulative distribution function(c.d.f.) of X and Y then it lies
in the interval…
A) [–1, 0] B) [0, 1]
C) [–1, 1] D) [0, –1]
35) For which of the following conditional expectations, the regression line of Y on X is
linear ?
A) E(Y|X = x ) = 1/ X B) E(Y|X = x ) = 0.2 X + 3
C) E(Y|X = x ) = 1/ Y D) none of these
36) If f(x) = 1, 5 < x < 6 and f(y) = 1, 3 < x < 4 then ------------
A) E(X) < E(Y) B) E(X) > E(Y)
C) E(X) = E(Y) D) none of these
37) If m.g.f. of independent continuous r.v.s X and Y is same and it is M(t) then m.g.f. of
a r.v. X+Y is…
A) 2M(t) B) 0
C) 1 D) M(t).M(t)
38) If X and Y are two independent continuous r. v.’s then …
A) Covariance (X,Y) = 0 B) Correlation(X,Y) = 0
C) E(XY) = E(X).E(Y) D) All of these
39) For bivariate continuous r.v. (X,Y) which of the following is not true ?
A) Cov(X,Y) = Cov(Y, X) B) Cov(–X, –X) = Cov(X, X)
C) Cov(X,3) = Cov(3, Y) D) Cov(–X, –Y) = – Cov(X, Y)
40) For the following joint p.d.f. of bivariate continuous r.v. (X,Y) the value of c is -----
c 0  x  y  1
f ( x) = 
0 otherwise
A) 1 B) 4
C) 2 D) 1/3

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41) If Var(X) = 1, Var(Y) = 9 and Cov(X, Y) = 1 then r(X, Y) is…
A) 1/3 B) 0
C) – 1 D) – 1/3
42) If E(Y|X = x ) = X , then regression coefficient of Y on X is…
A) 0.5 B) 0.1
C) 1 D) 0
 
43) If E E ( X / Y ) = 5 then …
A) E(X) = 5 B) E(Y) = 5
C) V(Y) = 5 D) V(X) = 5
44) If Joint p.d.f. of X and Y is f(x,y) = 3 – x – y; 0 < x < 1 ; 0 < y < 1 then marginal
distribution of y is….
A) f(y) = 2.5 – y B) f(y) = y – 2.5
C) f(y) = 3 – y D) f(y) = 3
45) For the following joint p.d.f. of bivariate continuous r.v. (X, Y) the value of k is…
8𝑘𝑥𝑦 0 < 𝑥 < 1, 0 < 𝑦 < 1
𝑓(𝑥) = {
0 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒

A) 1 B) 4
C) 2 D) 1/2
46) Let (X, Y) be the bivariate continuous random variable has following joint pdf
f (x, y) = 1 ; 0 < x < 1; 0 < y < 1
=0 ; Otherwise
Then P {0 < X < 0.5; 0.5 < Y < 1} will be
A) 0.25 B) 0.50
C) 0.75 D) 1
47) IF X and Y are independent continuous r.v.s, then…
A) E (Y|X) = E(X) B) E(X|Y) =E(Y)
C) V(Y|X) = V(X) D) E(Y|X) =E(Y)
48) If r.v. X has p.d.f. f(x) = 3x2 , 0 < x < 1 then range of variable Y = 2X+3 is…….
A) (0,2) B) (2, 3)
C) (3,5) D) (0, 1)
49) If r.v. X has p.d.f. f(x) = 3x2 / 2, – 1< x < 1 then range of variable Y = X2 is…….
A) (0,2) B) (–1, 1)
C) (0,1) D) (0, 3/2)

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50) If (X,Y) be the bivariate continuous r.v.s with joint p.d.f. f(x,y) then joint p.d.f. of
U=g1(x,y) and V=g2(x,y) is g(u,v) =……..
A) f(x).f(y) where x and y are in terms of u and v
B) f(x,y) where x and y are in terms of u and v
C) f(x,y)|J| where x and y are in terms of u and v
D) None of these.
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Long Answer Questions (10 Mark each)

1) Define Poisson distribution and find its mean and variance.

2) Define Poisson distribution. Find it’s p. g. f., mean and variance.
3) Show that under certain conditions to be stated, Poisson distribution is limiting
case of Binomial distribution
4) Define Negative binomial distribution and find its mean and variance.
5) Define Geometric distribution and find its mean and variance
6) For a univariate continuous r.v. X. Define
i) Mean ii) Mode iii) First Quartile
iv) Moment generating function v) rth central moment
7) For a univariate continuous r.v. X. Define
i) Probability density function ii) Median iii) Harmonic mean
iv) Cumulant generating function v) rth order raw moment
8) A continuous r. v. X has following p.d.f.
C.x(2 − x) 0  x  2
f ( x) = 
 0 otherwise
Find i) C ii) mean iii) variance iv) Mode
9) For a bivariate continuous r. v. (X, Y) show that
i) E(X +Y) = E(X) +E(Y)
ii) If X and Y are independent then E(XY) = E(X)E(Y)
10) A bivariate r. v. (X, Y) has joint pdf
3 − x − y 0  x  1, 0  y  1
f ( x, y ) = 
 0 otherwise
i) Find marginal distribution of X and Y
ii) Find E(X), E(Y) iii) E(XY) and Cov(X,Y)

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11) A bivariate r. v. (X, Y) has joint pdf

 xe − x (1+ y ) x  0, y  0
f ( x, y ) = 
 0 otherwise
i) Find marginal p.d.f. of X and Y
ii) Find conditional p.d.f. of Y given X and E(Y / X)
iii) State whether the regression of Y on X is linear or not.

12) A bivariate r. v. (X, Y) has joint pdf

4 x(1 − y ) 0  x  1, 0  y  1
f ( x, y ) = 
 0 otherwise
i) Find marginal p.d.f. of X and Y
ii) Find conditional p.d.f. of X given Y and E(X / Y)
iii) Are X and Y are independent ?

13) The joint p. d. f of (X,Y) is

𝐾xy 0 < 𝑥 < 1, 0 < 𝑦 < 1
f(x, y) = {
0 𝑜. 𝑤.
i) Find the value of constant K.
ii) Check whether X and Y are independent?
iv) Obtain E(X) and V(Y).

14) A bivariate r. v. (X, Y) has joint pdf

1 0  x  1, 0  y  1
f ( x, y ) = 
0 otherwise
Find the distribution of U = XY

15) A bivariate r. v. (X, Y) has joint pdf

−(𝑥+𝑦)
𝑓(𝑥, 𝑦) = {𝑒 0 ≤ 𝑥 < ∞, 0 ≤ 𝑦 < ∞
0 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
Find p.d.f. of U = (X +Y)/2

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 Short Answer Questions (5 Mark each)

1) Define Poisson distribution and find its recurrence relation for probabilities.
2) State and prove additive property of Poisson distribution
3) If X and Y are two independent Poisson variates with parameter 2 and 3
respectively, Find P(X+Y < 2)
4) Define Geometric and find its recurrence relation for probabilities.
5) State and prove lack of memory property of geometric distribution.
6) Define Geometric distribution and find its p.g.f.
7) Define Geometric distribution and find its cumulative distribution function(c.d.f.)
8) Define Negative binomial distribution and find its recurrence relation for
probabilities.
9) Find mean of Negative binomial distribution
10) Define moment generating function (m. g. f.) for univariate continuous r. v. and
explain how to obtain raw moments from m.g.f.
11) Define cumulative distribution function (c.d.f.) of continuous univariate random
variable and state its properties.
12) Define central moments and cumulants for continuous univariate random variable
and state relation between them up to order four
13) Define moment generating function (m.g.f.) and state their properties
14) For a univariate continuous r.v. X. Define
i) Mean ii) Mode iii) First and Third Quartile
15) The following is the p.d.f of continuous r.v. X
3
 x(2 − x) 0  x  2
f ( x) =  4

 0 otherwise

Find Mode of the r.v. X

16) Define probability density function (p.d.f) of continuous r.v. X and find mean for
the following pdf.
3
 x(2 − x) 0  x  2
f ( x) =  4

 0 otherwise

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17) Define probability density function (p.d.f) of continuous r.v. X and verify
following function is pdf.
 x2 0  x 1

f ( x) =  x(2 − x) 1  x  2
 0 otherwise

18) For a bivariate continuous r.v.(X, Y), Show that E(X–Y)= E(X) – E(Y)
19) Define joint cumulative distribution function for a bivariate continuous r.v. (X,Y)
and state their properties.
20) For a bivariate continuous r.v.(X, Y).
Define i) Conditional expectation ii) Conditional variance
21) For Bivariate continuous r. v. s (X, Y) define the terms
i) Marginal distribution of X.
ii) Conditional distribution of X given Y=y.
22) Let (X, Y) be continuous r. v., prove that E{E(X/Y)}=E(X).
23) The joint p.d.f of (X,Y) is
1 / 4 − 1  x, y  1
f ( x, y ) = 
 0 o.w

Then find i) P(X > 0 , Y > 0) ii) P(X >1/2 )

24) A bivariate r. v. (X, Y) has joint pdf
1 0  x  1, 0  y  1
f ( x, y ) = 
0 otherwise
Find marginal p.d.f. of X and Y
25) A continuous r. v. X has p.d.f.
x
 2 x4
f ( x) =  6
 0 otherwise

Find the probability distribution of Y = X/2

26) A continuous r. v. X has p.d.f.
2 x 0  x  1
f ( x) = 
 0 otherwise
Find the probability distribution of Y = 3X+1

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27) Let X be a r. v having p. d. f.
1
−1 ≤𝑥 ≤ 1
f(x)={2
0 𝑜. 𝑤.
If Y=X2 then obtain i) p. d. f of Y ii) E(Y)
28) Let X be a r.v having p.d.f.
1 0≤𝑥≤1
f(x)={
0 𝑜. 𝑤.
If Y = 1 – X then obtain i) p. d. f of Y ii) E(Y)
29) A continuous r. v. X has p.d.f.
1
 −3 x  3
f ( x) =  6
 0 otherwise

Find the probability distribution of Y = X2

30) A continuous r. v. X has p.d.f.
1 / 2 − 1  x  1
f ( x) = 
 0 otherwise
Find the probability distribution of Y = X2
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