2014 Question Paper, Solapur University Question Paper,B.Sc. II (Semester – III),STATISTICS (Paper – V)

Solapur University Question Paper
B.Sc. II (Semester – III) Examination, 2014
STATISTICS (Paper – V)
Continuous Probability Distributions – I
Day and Date : Tuesday, 27-5-2014 Max. Marks : 50
Time : 11.00 a.m. to 1.00 p.m.
Instructions : 1) All questions are compulsory and carry equal marks.
2) Figures to the right indicate full marks.
1. Choose the correct alternative : 10
1) If X is a random variable having it p.d.f. f(x), the E(X) is called
a) arithmetic mean b) geometric mean
c) harmonic mean d) none of these
2) Two random variables X and Y are said to be independent ; if
a) E(XY) = 1 b) E(XY) = 0
c) E(XY) = E(X)E(Y) d) None of these
3) If FX(x) is the cumulative distribution function (cdf) of a continuous r.v.X, then
it is
a) decreasing function of X b) non-decreasing function of X
c) both a and b d) none of these
4) If M is the median of continuous r.v.X with p.d.f. f(x), then ∫
−∞
M
f(x)dx will be
equal to
a) 1 b) 0 c) 2
1 d) none of these
5) M.G.F. of sum of independent r.v.’s is equal to
a) sum of their m.g.f. b) product of their m.g.f.
c) both a and b d) none of these
P.T.O.
Seat
No.
SLR-C – 61 -2- 
6) If X and Y are independent r.v.’s then the c.d.f FX,Y(x, y) is equal to
a) F (x) F (y) X ⋅ Y b) P(X ≤ x)⋅P(Y ≤ y)
c) both a and b d) none of these
7) If X~ U(0,1), then E(X) is equal to
a) 0 b) 1 c) 2
1 d) none of these
8) The distribution function of a continuous uniform distribution of a variable X
lying in the interval (a, b) is
a) b a
1
− b) b a
X a
−
− c) X a
b a
−
− d) none of these
9) If X ~ exp(θ), then the Var(X) will be equal to
a) θ
1 b) 2
1
θ c) θ d) none of these
10) Memoryless property holds in case of
a) uniform distribution b) exponential distribution
c) both a and b d) none of these
2. Answer any five of the following. 10
For a continuous random vector (X, Y), define :
i) Marginal p.d.f. of X and Y
ii) Conditional distribution of X given Y = y
iii) Expectation of a function g(X, Y)
iv) Conditional expectation of X given Y = y
v) Conditional variance of X given Y = y
vi) Cov (X, Y).
3. A) Answer any two of the following. 6
i) Let the r.v.X with p.d.f. f(x) given by
= ω
= ≤ ≤
0 ; 0
f(x) kx; 0 x 1
Find k and mean of X.
ii) For given the joint p.d.f. of (X, Y)
= ω
= ≤ ≤ ≤ ≤
0 ; 0
y ; 0 x 2, 0 y 1 2
3 f(x,y) 2
Are X and Y are independent ?
iii) If
= ω
π ≤ ≤ − π
π =
0 ; 0
2 x 2 ;
1 f(x)
Find the p.d.f. of Y = tanx.
B) State and prove the multiplication theorem of expectation. 4
4. Answer any two of the following. 10
i) The p.d.f. of a continuous r.v.X is given by
= ω
= ≤ ≤
0 ; 0
f(x) 3x ; 0 x 1 2
Find mean and vraince of X.
ii) Let X and Y be continuous r.v.’s having joint p.d.f.

= ω
< <
= − < <
0 ; 0
0 y 1
f (x, y) 12 xy(1 y) ; 0 x 1
Show that X and Y are independent.
iii) Define uniform distribution over (a, b). Obtain the variance of distribution.
5. Answer any two of the following. 10
i) Probability density function (p.d.f) of r.v.X is given by
= ω
= < <
0 ; 0
; 0 x 2 2
x f(x)
Find variance and median of X.
ii) If X has uniform distribution over (0, 1). Find the distribution of Y = –2logeX.
iii) If X~exp(θ), then find its m.g.f. and hence E(X).
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