Show that by expanding e^tXin a power series and taking expectations term by term we may write the momentgenerating function as

Thus, the coefficient of t^r / r! in this expansion is μ_r€², the rth origin moment. Write the power series expansion for M_X (t) for a gamma random variable and determine μ₁€² and μ₂€² using this approach.

Transcribed Image Text:

Mx (t) = E(e* ) =1+µ{t + µ - +- 2! +H,-+.. r!