Question 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.
Question 2) Explain joint probability distribution function and give its properties.
Question 3) Explain cumulants and mention its utilities.
Question 4) Describe MGF and give its properties.
Question 5) Explain Poisson distribution. Illustrate that the difference of two Poisson variates is not a Poisson variate.
Question 6) Define Negative Binomial distribution and obtain its probability generating function.
Question 7) Write down the characteristics of normal distribution?
Question 8) Write down the cumulant generating function of Gamma distribution.
Question 9) Define Chi-square statistics and give its uses.
Question 10) Establish the relation between F and t statistics.
Question 11)a) Define probability generating function and give its properties. Obtain its rotation with moment generating function.
Question 12)a) Show that the linear combination of independent normal variables is also a normal variable.
b) Explain Beta distribution of second kind and determine its mean and variance.