University Of Pune Question Paper
T.Y. B.A. Examination, 2010
STATISTICS (Special)
(Distribution Theory)
Time: 3 Hours Max. Marks: 80
N.B. : 1) All questions are compulsory.
2) Figures to the right indicate full marks.
3) Use of calculator and statistical table is allowed.
4) Symbols and abbreviations have their usual meanings.
1. Attempt any ten of the following : (2 each)
a) State the relation between normal and lognormal distribution.
b) State Chebyshev’s theorem for discrete random variable.
c) If → UX (–1, 1), state mean and variance of X.
d) State additive property of Cauchy’s distribution.
e) State relationship between Weibull and gamma distribution.
f) Define order statistics and state where it can be used.
g) If X →L (0, 1). State its distribution function (d.f.)
h) Give real life situation where truncated binomial distribution can be used.
i) If (X, Y) → BN (0, 0, 1, 4, 0.5) then state conditional distribution of Y given
X = 1.
j) If ( ) 2
1 → BX 100, then compute p (X < 50) by CLT.
k) If X → C (0, 1) then state the distribution of X
1 .
l) If (X1, X2, X3) → M D (n, p1, p2, p3) where i nX
3
11
=∑= and 1pi
3
11
=∑= then state
the distribution of X2.
m) If X → ( ) 22 ππ− /,/U then define its p.d.f and d.f. of X.
P.T.O.
[3801] – 376 -2-
2. Attempt any two of the following : (5 each)
a) Let X → U (a, b), find its mean deviation about mean.
b) If X → W (α , β ) then find its median.
c) State and prove Chebyshev’s inequality for continuous random variable.
d) Let (X1 X2 ---- Xk) → M D (n; p1, p2, ---- pk) where ∑pi = 1 and
Xi n n
11
=∑= then find its m.g.f. and hence obtain marginal distribution of X1.
3. Attempt any two of the following : (10 each)
a) State and prove Central Limit Theorem (CLT) for i, i, d r.v.s. based on m.g.f.
b) If X → L (μ , λ ) then obtain its first four cumulants and hence find β 1 and
β 2.
c) If X and Y are two independent U (0, 1) variates then find the distribution of
(X – Y) also calculate P [ |X – Y| < 0.5].
d) Let (X, Y) → B N (0, 0, 1, 1, ρ). Derive the distribution of u = Y
X
and
identify the distribution.
4. Attempt any two of the following : (15 each)
a) i) Let X have Poisson distribution with parameter ‘m’. If the distribution is
left truncated at X = 0 (value zero not observable) find p.m.f. of the resulting
distribution. Also find its mean and variance. 8
ii) If X → L N (a, μ , σ 2), find its rth moment about ‘a’ and hence find
mean. 7
b) i) A continuous random variable X has Laplace distribution with parameter
μ = 2 and λ =
2
3
find p(|X|< 5) and p(–3 < X < 1). 8
ii) If (X1, X2, ----- Xk) → MD (n; p1, p2, -- pk) then find Corr (Xi
, Xj
) 7
c) i) If X1, X2, .... Xn is a random sample from exponential distribution with
mean θ . Derive the distribution of X(n).
Where X(n) = Max. {x1, x2 ... xn}. 8
ii) If X → C (μ, λ ) find its Q1 and Q3. 7
d) i) If X and Y are two independent N (0, 1) variates, obtain the distribution
of
Y
X . 8
ii) A continuous r.v. X such that E (X) = 7 and E (X2) = 53. What is the least
value of p (3 < X < 11) ? and what is the value of ‘K’ such that
p [ |X –7 | ≤ K] ≥ 0.96 ? 7
_____________
T.Y. B.A. Examination, 2010
STATISTICS (Special)
(Distribution Theory)
Time: 3 Hours Max. Marks: 80
N.B. : 1) All questions are compulsory.
2) Figures to the right indicate full marks.
3) Use of calculator and statistical table is allowed.
4) Symbols and abbreviations have their usual meanings.
1. Attempt any ten of the following : (2 each)
a) State the relation between normal and lognormal distribution.
b) State Chebyshev’s theorem for discrete random variable.
c) If → UX (–1, 1), state mean and variance of X.
d) State additive property of Cauchy’s distribution.
e) State relationship between Weibull and gamma distribution.
f) Define order statistics and state where it can be used.
g) If X →L (0, 1). State its distribution function (d.f.)
h) Give real life situation where truncated binomial distribution can be used.
i) If (X, Y) → BN (0, 0, 1, 4, 0.5) then state conditional distribution of Y given
X = 1.
j) If ( ) 2
1 → BX 100, then compute p (X < 50) by CLT.
k) If X → C (0, 1) then state the distribution of X
1 .
l) If (X1, X2, X3) → M D (n, p1, p2, p3) where i nX
3
11
=∑= and 1pi
3
11
=∑= then state
the distribution of X2.
m) If X → ( ) 22 ππ− /,/U then define its p.d.f and d.f. of X.
P.T.O.
[3801] – 376 -2-
2. Attempt any two of the following : (5 each)
a) Let X → U (a, b), find its mean deviation about mean.
b) If X → W (α , β ) then find its median.
c) State and prove Chebyshev’s inequality for continuous random variable.
d) Let (X1 X2 ---- Xk) → M D (n; p1, p2, ---- pk) where ∑pi = 1 and
Xi n n
11
=∑= then find its m.g.f. and hence obtain marginal distribution of X1.
3. Attempt any two of the following : (10 each)
a) State and prove Central Limit Theorem (CLT) for i, i, d r.v.s. based on m.g.f.
b) If X → L (μ , λ ) then obtain its first four cumulants and hence find β 1 and
β 2.
c) If X and Y are two independent U (0, 1) variates then find the distribution of
(X – Y) also calculate P [ |X – Y| < 0.5].
d) Let (X, Y) → B N (0, 0, 1, 1, ρ). Derive the distribution of u = Y
X
and
identify the distribution.
4. Attempt any two of the following : (15 each)
a) i) Let X have Poisson distribution with parameter ‘m’. If the distribution is
left truncated at X = 0 (value zero not observable) find p.m.f. of the resulting
distribution. Also find its mean and variance. 8
ii) If X → L N (a, μ , σ 2), find its rth moment about ‘a’ and hence find
mean. 7
b) i) A continuous random variable X has Laplace distribution with parameter
μ = 2 and λ =
2
3
find p(|X|< 5) and p(–3 < X < 1). 8
ii) If (X1, X2, ----- Xk) → MD (n; p1, p2, -- pk) then find Corr (Xi
, Xj
) 7
c) i) If X1, X2, .... Xn is a random sample from exponential distribution with
mean θ . Derive the distribution of X(n).
Where X(n) = Max. {x1, x2 ... xn}. 8
ii) If X → C (μ, λ ) find its Q1 and Q3. 7
d) i) If X and Y are two independent N (0, 1) variates, obtain the distribution
of
Y
X . 8
ii) A continuous r.v. X such that E (X) = 7 and E (X2) = 53. What is the least
value of p (3 < X < 11) ? and what is the value of ‘K’ such that
p [ |X –7 | ≤ K] ≥ 0.96 ? 7
_____________
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