problem 1) Prove that the Geometric mean G of the distribution:

dF = 6(2-x) (x-1) dx,  1 ≤ x ≤ 2 is given by 6 log (16G) = 19.

problem 2) describe joint probability distribution function and give its properties.

problem 3) describe cumulants and mention its utilities.

problem 4) Describe MGF and give its properties.

problem 5) describe Poisson distribution. Illustrate that the difference of two Poisson variates is not a Poisson variate.

problem 6) Define Negative Binomial distribution and obtain its probability generating function.

problem 7) prepare down the characteristics of normal distribution?

problem 8) prepare down the cumulant generating function of Gamma distribution.

problem 9) Define Chi-square statistics and give its uses.

problem 10) Establish the relation between F and t statistics.

problem 11)a)  Define probability generating function and give its properties. Obtain its rotation with moment generating function.

problem 12)a) Show that the linear combination of independent normal variables is also a normal variable.

b)  describe Beta distribution of second kind and determine its mean and variance.