problem 1:

a) Define the term Laplace distribution. Obtain its characteristic function. And hence obtain its mean and variance.

b) Define Weibull distribution. Obtain its characteristic function. And hence obtain its mean and variance.

problem 2:

a) Define the term log-normal distribution. State and prove its reproductive property.

b) Define the term logistic distribution and obtain its m.g.f.

problem 3:

a) Derive the distribution of non-central chi-square.

b) Derive the joint distribution of the jth and kth order statistic for 1 ≤ j < k ≤ n.

problem 4:

a) Derive the distribution of non-central F and therefore show that central F is a particular case of it.

b) Derive the distribution of central t.

problem 5: prepare brief notes on any two of the given terms:

a) Chebyshev’s inequality.
b) Kolomogorov’s strong law of large numbers.
c) Compound binomial distribution.
d) Interrelationships between chi-square, t and F.
e) Cauchy Criterion.