- #1

sanctifier

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## Homework Statement

Question : Prove [itex] \varphi (t) = \sum_{i=0}^{ \infty } a_icos(it) [/itex] is a moment generating function (m.g.f.) and determine its corresponding probability density function (p.d.f.)

when [itex] \sum_{i=0}^{ \infty } a_i=1 [/itex] holds for [itex] a_i \geq 0 [/itex].

## Homework Equations

Nothing special.

## The Attempt at a Solution

I really don't know what to do with this question, all I know is [itex] \varphi (0) = \sum_i p(x_i) = 1 [/itex] in the discrete case.

For the current one,

[itex] \varphi (0) = \sum_{i=0}^{ \infty } a_i = 1 [/itex]

This is a proof? It cannot be so easy, and how to determine the p.d.f. when its m.g.f. is given?

Thank you in advance!