Probability-Moment Generating Functions [ISS (Statistical Services) Statistics Paper I (Old Subjective Pattern)]: Questions 1 - 1 of 1

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Question number: 1

» Probability » Moment Generating Functions

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Appeared in Year: 2014

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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.

Explanation

The moment generating function of S N is where S N =X 1 +X 2 +…+X N, X i are i. i. d random variable and N is also a random variable.

X i ‘s are independent identically distributed, so

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