# LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034.

B.Sc. DEGREE EXAMINATION – MATHEMATICS

FourTH SEMESTER – APRIL 2003

## ST 4201 / sTA 201 – MATHEMATICAL STATISTICS

28.04.2003

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 |

P (X=x) | 0 | k | 2k | 2k | 3k | k^{2} |
2k^{2} |
7k^{2}+k |

Find k.

- Let f (x) =

0 ; else where

Find E(X).

- 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

f(x) =

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,s
^{2}). i) Find b so that - ii) If P (X < 89) =0.90 and P(X < 94) =0.95, find m and s
^{2}. - 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 x
_{1}and x_{2}have the joint p.d.f

f (x_{1 ,}x_{2}) =

i ) find the conditional mean of X_{1} given X_{2} and ii) the correlation coefficient

between X_{1} and X_{2.}

- 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

in normal.

- a) Derive the p.d.f of F- variate with (n
_{1},n_{2}) d.f. - Find the g.f of exponential distribution.

- a) Let X
_{1}, X_{2}, …. X_{n}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.