Total No. of Questions 10] [Total No. of Pages 02
M.Sc. DEGREE EXAMINATION, MAY 2017
First Year
STATISTICS
Probability and Distribution Theory
Time 3 Hours Maximum Marks: 70
Answer any Five questions.
All questions carry equal Marks.
Q1) Define distribution function. State and prove its properties.
State and prove Kolmogorov's inequality.
Q2) State and prove a necessary and sufficient condition for n random variables
to be independent.
State and prove Borel-Cantelli lemma.
Q3) Explain modes of convergence. In the usual notation, prove
State and prove Kolmogorov's strong law of large numbers for
independent random variables.
Q4) State and prove Levy and Lindberg form of central limit theorem.
Determine whether strong law of large numbers holds for the sequence of
random variables
Q5) Derive compound binomial distribution.
Define multinomial distribution. Show that the marginal p.m.f. of each
Xi, i …… k−1 in a multinomial distribution is binomial.
Q6) Derive compound Poisson distribution.
Q7) Obtain for a Weibull random variable Y.
Define log-normal distribution. Obtain its nth row moment.
Q8) Define Laplace distribution. Obtain its m.g.f.
Define logistic distribution. Obtain its characteristic function.
Q9) Derive the distribution of t.
Derive the joint p.d.f. of
Q10) Derive the distribution of non-central Chi-square.
Obtain the joint p.d.f. of and 1 j k n.