Question

(10 points) Consider a discrete random variable X, which can only take on non-negative integer values, with E[Xk] = 0.8, k = 1,2, .... Use the moment generating function approach to find the pmf of Px(k), k = 0,1,....

Answer 1

Answer:

Given that:

Consider a discrete random variable X, which can only take on non-negative integer values with E[X^k] = 0.8, k=1,2... Use the moment generating function approach to find the pmf of $p_x(k)$ , k=0,1,...

here as mgf

My(t) = 1+ E(X) *t/1! + E(X)*+/2! + E(X3) * t/3! + ..............

= 0.2 +0.8 +0.8t/1! +0.8t/2! +0.8ť/3! + .......

= 0.2 +0.8*(1+t/1!+t/2!+ t/3! + ....)

= 0.2 +0.8e

comparing it with mgf of discrete pmf : $\sum xe^{tx}$

below is pmf of X:

P(X=0) =0.2

P(X=1) =0.8

anywhere else P(x) =0