LOYOLA COLLEGE (AUTONOMOUS), CHENNAI –600 034
M.Sc., DEGREE EXAMINATION – MATHEMATICS
THIRD SEMESTER – NOVEMBER 2004
ST 3951 – MATHEMATICAL STATISTICS – I
02.11.2004 Max:100 marks
1.00 – 4.00 p.m.
SECTION – A
Answer ALL the questions (10 ´ 2 = 20 marks)
- Find C such that f (x) = C satisfies the conditions of being a pdf.
- Let a distribution function be given by
0 x < 0
F(x) = 0 £ x < 1
1 x ≥ 1
Find i) Pr ii) P [X = 0].
- Find the MGF of a random variable whose pdf is f (x) = , -1 < x < 2, zero elsewhere.
- If the MGF of a random variable is find Pr [X = 2].
- Define convergence in probability.
- Find the mode of a distribution of a random variable with pdf
f (x) = 12x2 (1 – x), 0 < x < 1.
- Define a measure of skewness and kurtosis using the moments.
- If A and B are independent events, show that AC and BC are independent.
- Show that E (X) = for a random variable with values 0, 1, 2, 3…
- Define partial correlation.
SECTION – B
Answer any FIVE questions. (5 ´ 8 = 40 marks)
- Show that the distribution function is non-decreasing and right continuous.
- ‘n’ different letters are placed at random in ‘n’ different envelopes. Find the probability that none of the letters occupies the envelope corresponding to it.
- Show that correlation coefficient lies between -1 and 1. Also show that p2 = 1 is a necessary and sufficient condition for P [Y = a + bx] = 1 to hold.
- Derive the MGF of gamma distribution and obtain its mean and variance.
- Let f (x, y) = 2 0 < x < y < 1 the pdf of X and Y. Obtain E [X | Y] and E [Y | X]. Also obtain the correlation coefficient between X and Y.
- Show that Binomial distribution tends to Poisson distribution under some conditions.
- State Chebyshw’s inequality. Prove Bernoulli’s weak law of large numbers.
- 4 distinct integers are chosen at random and without replacement from the first 10 positive integers. Let the random variable X be the next to the smallest of these 4 numbers. Find the pdf of X.
SECTION – C
Answer any TWO questions (2 ´ 20 = 40 marks)
- a) Let {An} be a decreasing sequence of events. Show that
P . Deduce the result for increasing sequence.
- b) A box contains M white and N – M red balls. A sample of size n is drawn from the
box. Obtain the probability distribution of the number of white balls if the sampling is
done i) with replacement ii) without replacement. (10+10)
- a) State any five properties of Normal distribution.
- In a distribution exactly Normal 7% are under 35 and 89% are under 63. What are the mean and standard deviation of the distribution?
- If X1 and X2 are independent N and Nrespectively, obtain the distribution of a1 X 1 + a2 X2. (5+10+5)
- a) Show that M (t1, t2) = M (t1, 0) M (0, t2) “, t1, t2 is a necessary and sufficient condition
for the independence of X1 and X2.
- b) Let X1 and X2 be independent r.v’s with
f1 (x1) = , 0 < x1 < ¥
f2 (x2) = , 0 < x2 < ¥
Obtain the joint pdf of Y1 = X1 + X2 and Y2 =
Also obtain the marginal distribution of Y1 and Y2
- c) Suppose E (XY) = E (X) E (Y). Does it imply X and Y are independent.
(6+10+4)
- a) State and prove Lindberg-Levy central limit theorem.
- b) Let Fn (x) be distribution function of the r.v Xn, n = 1,2,3… Show that the
sequence{Xn} is convergent in probability to O if and only if the sequence Fn (x)
satisfies
= 0 x < 0
1 x ≥ 0
- c) Let Xn, n = 1, 2, … be independent Poisson random variables. Let y100 = X1 + X2 + …+
X100. Find Pr [190 £ Y100 £ 210]. (8+8+4)
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