Probability (ISS Statistics Paper I (Old Subjective Pattern)): Questions 63 - 67 of 72

X ₁, X ₂, …, X _N are independently, identically distributed random variables. Define S _N = X ₁ + X ₂ + … + X _N , where N is a random variable independent of X _i, i = 1, 2, … N.

Show that the moment generating function (mgt) of S _N is

where My (t) is the mgf of a random variable Y. Hence find the mgf of S _N when N follows a Poisson distribution with parameter λ. and X _i follows an exponential distribution with mean parameter θ, i = 1 to N.

Show that the sum of two independent Poisson random variables with parameters λ and µ respectively is a Poisson random variable with parameter λ+µ.

The joint density of (X, Y) is

Find the conditional densities and E [X|Y = 1.5].

Let the joint p. d. f. of (X, Y) be f (x, y) = e ^-y, 0 < x < y < ∞.

Obtain the probability P (X + Y ≤ 1).

Find the generating function of X whose probability density function is

P [X = r] =pq ^r-1, r = 1, 2, …, 0 < p < 1, q = 1 - p