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6.
[Thi s qu esti o n pape r co ntain s 4 print ed pages. ]
E
Your Roll N o ............... .
Sr. N o. of Question Paper
I I 63
U niqu e Pa pe r Co d e
23 5 204
Na m e o f th e Co urse
B.Sc. (Hons.) Mathematics
Nam e o f th e Pa pe r
Prob ability & St ati stics - MAHT 203
Semeste r
II
Maximum Ma rks : 75
Du ra ti o n : 3 Ho urs
Instructions for Candidates
I.
Write your Roll No . on the top immediately on receipt of this question pap er.
2.
In all the re are six questions .
3.
Question No. 1 is compulsory and it contains five parts of 3 marks each.
4.
In Question No. 2 to 6, attempt any two parts from three parts. Each part
carries 6 marks .
5.
I.
Use of scientific calculator is allowed .
(i) If C and C are events in a sample space S. Then prove that
2
1
P(C1 vc2)
=
P(C1)+ P(C2)-P(C1nC2) .
(ii) A bowl contains 16 chips, of which 6 are red, 7 are white and 3 are blue.
If four chips are taken at random and without replacement, find the probability
that : (a) each of the 4 chips is red; (b) none of the 4 chips is red ; ( c) there
is at least I chip of each color.
(iii) Let X and X have the joint pdf f(xl'x 2) = x 1 + x 2, 0 < x 1 < x 2 < I, zero
2
1
elsewhere. Find the marginal pdfs of X 1 and X 2.
P.TO.
2
1163
(iv) Show that if a random variable has a uniform density with the parameters a
a nd
p,
the probability that it will take on a value less than a+ p(P - a) is
equal to p.
(v) If X is a random variable with E(X) = 3 and E(X 2 ) = 13 . Use the Chebyshev
ineq ual ity to determine a low er bound for P(-2 < X < 8).
2.
(i) Let {CJ be an arbitrary sequence ofevents. Then prove that
00
P(LJC.J ::;
n=I
I :=,P( en).
(ii) If X is a random variable having the Standard Normal distribution and
Y = X 1 , show that Cov(X, Y) = 0 even though X and Y are evidently not
independent.
(iii) Find the moment generating function of the gamma distribution and hence
find the mean and the variance.
3.
(i) State Negative Binomial distribution and find its mean and variance.
(ii) Show that if X is a random variable having the Poisson distribution with
parameter A and A
oo, then the moment generating function of
z=
x- A
'
that is, that of a standardized Poisson random variable, approaches the
moment generating function of the standard normal distribution.
(iii) Find the mean and the variance of the exponential distribution .
4.
(i) (a) If the probability is 0.75 that an applicant for a driver's license will pass
· . the road t~st on any given try, what is the probability that an applicant
will finally pass the test on the fourth try ?
3
1163
(b) If X has an exponential distribution , show that
P(X
t
+ TIX
(ii) Given the joint density f(x, y)
T)
= P(X
= 6x,
t).
0 < x < y < I, zero elsewhere. Find
(iii) If (X, Y) has a bivariate normal distribution, find the marginal distribution of
X and Y. Under what conditions X and Y are independent?
5.
(i) If the regression ofY on Xis linear, then show that
cou
(ii) (a) Let X be a random variable such that P(X :s; 0)
P[x 7 a.}'
/
-
2µ) :s;
,......e R"Ms. Show that P(X
i-
=
0 and let µ = E(X)
v 1--5 , ,,,_ e[ k.<lt.-4t,
f lx~).. K) &:... · 1\..{_ · _ ::;, / V
I
;)....k
--
Cb) Find the cdf ofa random variable X having pdff(x) = 6x(l - x), 0 < x < 1,
zero elsewhere.
(iii) Suppose X, and X 2 are random variables of the discrete type which have the
joint pmf
p ( x"x 2 )
x 1 + 2x 2 (
)
= -'---=-,
x"x 2
18
_
-
(1,1), (1,2), (2,1), (2,2), zero elsewhere.
Determine the mean and the variance of X 2 giYen X 1 = x 1 for x,
= l or 2.
Also compute E(3X 1 - 2XJ
6.
(i) State and Prove Chapman-Kolmogorov's equations.
P. T.O.
)--
l 163
4
(ii) Suppose that the chance of rain tomorrow depends on previous weather
conditions only through whether or not it is raining today and not on
pa st weather conditions. Suppose also that if it rains today, then it will
rain tomorrow with the probability 0.7; and if it does not rain today then
it will rain tomorrow with the probability 0.4 . Express this model as a
Markov chain and find its transition probability matrix. Calculate
the probability that it will rain four days from today given that it is raining
today.
(iii) State and prove Chebyshev 's Theorem.