Moment-Generating Function of the Sum of the Random Variables in a Random Sample

Topics: Statistics - Random Sample - Moment-Generating Function

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(proposition)

Let $X_1,\dots ,X_n$ be a random sample with a moment-generating function $M_{X_1}(t),\dots ,M_{X_n}(t)$.

Let:

$Y=a_1{X}_1+\cdots +a_n{X}_n$

…where $a_1,\dots ,a_n$ are constants. In that case, then:

$$M_Y(t)=\left(M_{X_1}(a_{1}t)\right)(\dots )\left(M_{X_n}(a_{n}t)\right)$$