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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI 600 034
B.Sc. DEGREE EXAMINATION MATHS.,ADV.ZOO., PLANT BIO., &PHYSICS
FOURTHSEMESTER APRIL 2018
ST 4209/ST 4206/ST 4201- MATHEMATICAL STATISTICS
Date: 02-05-2018 Dept. No. Max. 100 Marks
Time: 09:00-12:00
SECTION A
Answer ALL questions: (10X 2 20 Marks)
1. Define: Statistics.
2. What is conditional distribution?
3. Write the additive property of Binomial distribution.
4. How will you derive the marginal density function from joint density function?
5. Write the MGF of Poisson distribution.
6. Derive the mean of Exponential distribution.
7. What is the nth order statistic?
8. Define: t statistic.
9. Define: unbiased estimator.
10. Define: Type II error.
SECTION B
Answer any FIVE questions: X 8 40 Marks)
11. State and prove the addition law of probability.
12. If the joint pdf of is given by 0 f x y e x y x y . Find E
13. State and prove Chebyshev's inequality.
14. Calculate the correlation co efficient for the following data.
15. Prove that a linear combination of random variables X1, X2,…,Xn follow
is also Normal.
16. Derive the Mean and variance of Discrete Uniform distribution.
17. A random sample(X1, X2, X3, X4, X5) of size 5 is drawn from normal population with unknown
mean Consider the following estimators.

5
1 2 3 4 5
1
X X X X X
t

ii) 3
1 2
2 2
X
X X
t

iii)
3
2 1 2 3
3
X X X
t


Find λ. Are t1 and t2 unbiased? State giving reasons, the estimator which is best among t1, t2 and t3?
18. Define the following:
Null Hypothesis Alternate Hypothesis Critical region Most Powerful critical region
X 43 21 25 42 57 59
Y 99 65 79 75 87 81
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SECTION C
Answer any TWO questions X 20 40 Marks)
19. Two random variables X and Y have the joint pdf







otherwise
x y
xy
f x y
0
0 4,1 5
96 . Find
Var(Y) COV(X,Y).
20. Derive the moment generating function of Normal distribution. (10 Marks)
State and prove the lack of memory property of exponential distribution. (10 Marks)
21. Derive the moment generating function of chi square distribution and hence derive the mean and
variance.
22. State and prove Neyman Pearson Lemma.