LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034.
B.Sc. DEGREE EXAMINATION – MATHEMATICS
FourTH SEMESTER – APRIL 2003
ST 4201 / sTA 201 – MATHEMATICAL STATISTICS
9.00 – 12.00 Max : 100 Marks
PART – A (10´ 2=20 marks)
Answer ALL the questions.
- Two dice are thrown. What is the probability that the sum of the numbers on the two dice is eight?
- The probability that a customer will get a plumbing contract is and the probability that he will get an electric contract is 4/9. If the probability of getting at least one is 4/5,determine the probability that he will get both.
- Consider 2 events A and B such that and . Verify whether the given statement is true (or) false. .
- Define i) independent events and ii) mutually exclusive events.
- State any four properties of a distribution function.
- The random variable X has the following probability function
|X = x||0||1||2||3||4||5||6||7|
- Let f (x) =
0 ; else where
- Let X ~ B (2, p) and Y~B (4, p). If P , find P.
- Define consistent estimator.
- State Neyman – Pearson lemma.
PART – B (5´ 8=40 marks)
Answer any FIVE questions.
- A candidate is selected for three posts. For the first post three are three candidates, for the
second there are 4 and for the third there are 2. What are the chances of his getting
- i) at least one post and ii) exactly one post?
- Three boxes contain 1 white, 2 red, 3 green ; 2 white, 3 red, 1 green and 3 white, 1 red, 2 green balls. A box is chosen at random and from it 2 balls are drawn at random. The balls so drawn happen to be white and red. What is the probability that they have come from the second box?
- Find the conditional probability of getting five heads given that there are at least four heads, if a fair coin is tossed at random five independent times.
- Derive the mean and variance of hyper-geometric distribution.
- Let X be a random variable having the p.d.f
Find the m.g.f. of X and hence obtain the mean and variance of X.
- If X is B(n,p), show that E= p and E.
- Let X be N(m,s2). i) Find b so that
- ii) If P (X < 89) =0.90 and P(X < 94) =0.95, find m and s2.
- If X and Y are independent gamma variates with parameters m and n respectively,
Show that ~ .
PART – C (2´20=40 marks)
Answer any TWO questions
- If the random variables x1 and x2 have the joint p.d.f
f (x1 ,x2) =
i ) find the conditional mean of X1 given X2 and ii) the correlation coefficient
between X1 and X2.
- a) Find all the odd and even order moments of Normal distribution.
- Let (X,Y) have a bivariate normal distribution. Show that each marginal distribution
- a) Derive the p.d.f of F- variate with (n1,n2) d.f.
- Find the g.f of exponential distribution.
- a) Let X1, X2, …. Xn be a random sample of size n from N (q,1) . Show that the sample
mean is an unbiased estimator of the parameter q.
- Write a short note on:
- i) null hypothesis ii) type I and type II errors iii) standard error
- iv) one -sided and two -sided tests.