Moments

Class: STAT 131 Subject: probability Date: 2025-02-24 Teacher: **Prof. Marcela

Moment Generating Functions(MGF)

Geometric MGF

• For $X$ ~ $Geom(p)$, $M(t)=E(e^{X}t)=\frac{p}{1-e^{t}q}$
• for $qt$ < 1, i.e for t in $(-\infty ,log(1/q))$

Uniform MGF

• for $U$ ~ $Unif(a,b)$ $M(t)=E(e^{U}t)=\frac{e^{t}b-e^{t}a}{t(b-a)}$
• for $t\ne 0$ and $M(0)=1$

Sum of Independent R.V’s

• MGFs of a sum of independent r.v.s. If $X$ and $Y$ are independent, then the MGF of $X+Y$ is the product of the individual MGFs: $M_{X+Y}(t)=M_X(t)M_Y(t)$
• This is true because if $X$ and $Y$ are independent, then $E(e^t(X+Y)=E(e^{t}X)E(e^{t}Y))$

MGF of location-scale transformation.

• if $X$ has MGF $M(t)$, then the MGF of $a+bX$ is: