# 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

Appeared in Year: 2014

### Describe in Detail

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

In the