LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – STATISTICS

NO 19

FOURTH SEMESTER – APRIL 2008

ST 4501 – DISTRIBUTION THEORY

 

 

 

Date : 26/04/2008                Dept. No.                                        Max. : 100 Marks

Time : 9:00 – 12:00

 

SECTION – A

Answer ALL questions.:                                                                               (10 x 2 = 20)            

 

1. Explain the joint p.d.f of two continuous random variables X and Y.
2. Define conditional probability mass function.
3. Let . Find.
4. Write down any two properties of negative Binomial distribution.
5. Define Laplace distribution and find its mean.
6. Define Beta distribution of first kind.
7. Define students-t statistic and write down its probability density function.
8. State the additive property of Chi-square distribution.
9. Define order statistic and give an example.
10. Define conditional expectation and conditional variance of a random variable X given Y= y.

 

SECTION – B

Answer any FIVE  questions.                                                                      (5 x 8 = 40)

 

1. The joint p.d.f of random variables X and Y is given by

• Find the value of k
• Verify whether X and Y are independent.

1. Derive Poisson distribution as limiting form of Binomial distribution.
2. Define multinomial distribution and find the marginal distributions.
3. Explain joint distribution function of two dimensional random variable (X,Y) and establish any two of its properties..
4. Show that for normal distribution mean, median and mode coincide.
5. Find the MGF of Bivariate normal distribution.
6. State and prove central limit theorem.
7. Derive the p.d.f of F-distribution with n₁ and n₂ degrees of freedom.

 

SECTION – C

Answer any TWO   questions.                                                                      (2 x 20 = 40)

 

1. a) Obtain mean deviation about mean of Laplace distribution.

1. b) Show that exponential distribution satisfies lack of memory property.

1. a) Derive MGF of negative Binomial distribution and show that its mean is less than its variance.

1. b) Find the factorial moments of hyper-geometric distribution.

1. a) If X and Y are independent Chi-square variates with n₁ and n₂ d.f, find the p.d.f of x/x+y.

1. b) Obtain the MGF. of Binomial distribution with n=7 and p=0.6 and hence find .

1. a) Let X and Y follow Bivariate normal distribution with and P=0.4. Find the following probabilities.

(i)

(ii)

1. b) Let X₁, X₂, …. X_n be a random sample with common p.d.f

Find p.d.f, mean and variance of X₍₁₎, the first order statistic.

 

 

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