LOYOLA COLLEGE (AUTONOMOUS), CHENNAI– 600 034
B.Sc. DEGREE EXAMINATION – MATHEMATICS
Fourth SEMESTER – NOVEMBER 2003
ST 4201/STA 201 MATHEMATICAL STATISTICS
14.11.2003 Max: 100 Marks
9.00 – 12.00
SECTION – A
Answer ALL the questions. (10 ´ 2 = 20 Marks)
- Define an event and probability of an event.
- If A and B any two events, show that P (AÇBC) = P(A) – P(AÇB).
- State Baye’s theorem.
- Define Random variable and p.d.f of a random variable.
- State the properties of distribution function.
- Define marginal and conditional p.d.fs.
- Examine the validity of the given Statement “X is a Binomial variate with
mean 10 and S.D 4”.
- Find the d.f of exponential distribution.
- Define consistent estimator.
SECTION – B (5 ´ 8 = 40 Marks)
Answer any FIVE questions.
- An urn contains 6 red, 4 white and 5 black balls. 4 balls are drawn at random.
Find the probability that the sample contains at least one ball of each colour.
- Three persons A,B and C are simultaneously shooting. Probability of A hit the
target is ; that for B is and for C is Find i) the probability that
exactly one of them will hit the target ii) the probability that at least one of them
will hit the target.
- Let the random variable X have the p.d.f
Find P( ½ < X < ¾) and ii) P ( – ½ < X< ½).
- Find the median and mode of the distribution
.
- Find the m.g.f of Poisson distribution and hence obtain its mean and variance.
- If X and Y are two independent Gamma variates with parameters m and g
respectively, then show that Z = ~ b (m,g).
- Find the m.g.f of Normal distribution.
- Show that the conditional mean of Y given X is linear in X in the case of bivariate normal distribution.
SECTION – C
Answer any TWO questions. (2 ´ 20 = 40 Marks)
- Let X1and X2 be random variables having the joint p.d.f
Show that the conditional means are
(10+10)
- If f (X,Y) has a trinomial distribution, show that the correlations between
X and Y is .
- i) Derive the p.d.f of ‘t’ distribution with ‘n’ d.f
- ii) Find all odd order moments of Normal distribution. (15+5)
- i) Derive the p.d.f of ‘F’ variate with (n1,n2) d.f (14)
- ii) Define i) Null and alternative Hypotheses (2)
- ii) Type I and Type II errors. (2)
and iii) critical region (2)
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