LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – MATHEMATICS, PHYSICS & CHEMISTRY

AC 10

FOURTH SEMESTER – APRIL 2006

                                                 ST 4201 – MATHEMATICAL STATISTICS

(Also equivalent to STA  201)

 

 

Date & Time : 22-04-2006/9.00-12.00         Dept. No.                                                       Max. : 100 Marks

 

 

Part A

Answer all the questions.

1. Define conditional probability of the event A given that the event B has happened.
2. If A₁ and A₂ are independent events with P(A₁) = 0.6 and  P(A₂) = 0.3, find     P(A₁ U A₂), and P(A₁ U A₂^c)
3. State any two properties of a distribution function.
4. Define the covariance of any two random variables X and Y. What happens when they are independent?
5. The M.G.F of a random variable is  [(2/3) + (1/3) e^t]⁵ . Write the mean and variance.
6. Define a random sample.
7. Explain the likelihood function.
8. Let X have the p.d.f. f(x) =1/3, -1<x<2, zero elsewhere. Find the M.G.F.
9. Define measures of skewness and kurtosis through moments.
10. Define a sampling distribution.

 

Part B

Answer any five questions.

1. Stat and prove Bayes theorem.
2. Derive the mean and variance of Gamma distribution.
3. Let the random variables X and Y have the joint pdf

x + y, 0<x<1, 0<y<1

f(x, y) =

0, otherwise.

Find the correlation coefficient.

1. A bowl contains 16 chips of which 6 are red, 7 are white and 3 are blue. If 4 chips are taken at random and without replacement, find the probability that

1. All the 4 are red.
2. None of the 4 is red.

• There is atleast one of each colour.

1. State and prove the addition theorem for three events A, B and C. What happens when they are mutually exclusive?
2. Derive the mgf of Poisson distribution. And hence prove the additive property of the Poisson distribution.
3. Let X₁ and X₂ denote a random sample of size 2 from a distribution that is       N(m, s²). Let Y₁ = X₁ + X₂ , Y₂ = X₁ – X_2.  Find the joint pdf of Y₁ and Y₂ and show that Y₁ and Y₂ are independent.
4. Define the cumulative distribution function F(x) of a random variable X and mention the properties of it.

Part C

Answer any two questions.

19. a) Derive the recurrence relation for the central moments of Binomial distribution. Obtain the first four moments.
20. b) Show that Binomial distribution tends to poisson distribution under certain conditions.           (10 +10 = 20)
21. a) Discuss the properties of Normal distribution
22. b) In a distribution exactly normal, 10.03% of the items are under 25 kilogram weight and 89.97 % of the items are under 70 kilogram weight. What are the mean and standard deviation of the distribution?                                                                                                      (10 +10 = 20)
23. Let f(x, y) = 8xy, 0<x<y<1; f(x, y) = 0 elsewhere. Find
24. a) E(Y/X = x),    b). Var( Y/X = x).
25. b) If X and Y are independent Gamma variates with parameters m and v respectively, show that the variables U = X + Y, Z = X / (X + Y) are independent and that U  is a g( m + v) variate and Z is a b₁(m, v) variate.                                                                                      (10 +10 = 20)
26. a) Derive the pdf of t-distribution.
27. b) Obtain the Maximum Likelihood Estimators of m and s² for Normal distribution.         (10 + 10 = 20)

 

 

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