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  • STA256 Lecture 09

Moment Generating Function

• For full marks on a test, you must state the neighbourhood of the MGF.
• Discrete Random Variable
  • $M_x(t)=E[e^{tx}]$
  • $M_x(t)=\sum e^{t{x}_i}p(x_i)$
• Continuous Random Variable
  • $M_x(t)=E[e^{tx}]$
  • $M_x(t)={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{e}^{tx}f(x)dx$
• $k^{\text{th}}$ moment (Evaluated at $x=0$):
  • $M^{(k)}(0)=\frac{d}{d{t}^k}{M}_x(t)|{}_{t=0}=E[x^k]$
  • $k=1$
    • $M^1(0)=E(x^1)=\underset{\text{Mean}}{\underbrace{\mu }}$
  • $k=2$
    • $M^{″}(0)=E(x^2)$ is used to evaluate ${\sigma }^2=E(x^2)-{\mu }^2$
• Analogy to show how it characterizes distributions.
  • In MAT102 we can show $6$ and $8$ are even as they can be written in the form $n=2k$.
  • The MGF can do the same thing.

Examples

Discrete Random Variable

#tk in slides
Review finding $E[g(x)]$

Continuous Random Variable

• Let the random variable $X$ have a Probability Density Function
• $f(x)=2e^{-2x},x\ge 0$
• Find the Moment Generating Function of $X$, use the MGF to find the Mean and Variance.
• $M_x(t)=E[e^{tx}]$
• $M_x(t)={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{e}^{tx}f(x)dx$
• $M_x(t)={\int }_0^{\mathrm{\infty }}{e}^{tx}2e^{-2x}dx$
• $M_x(t)={\int }_0^{\mathrm{\infty }}2e^{(t-2)x}dx$
• $M_x(t)=\left[\frac{2e^{(t-2)x}}{t-2}\right]{}_0^{\mathrm{\infty }}$
• Examine $2e^{(t-2)x}$
  • For which $t$ does $2e^{(t-2)x}$ converge for $x\ge 0$?
  • $t=2$
    • $\underset{x\to \mathrm{\infty }}{lim}2e^{0x}=2$
  • Also for $t-2<0$
    • $t<2$
• $M_x(t)=\left[\frac{2e^{(t-2)x}}{t-2}\right]{}_0^{\mathrm{\infty }}$
• $M_x(t)=\underset{\begin{array}{c}x\to \mathrm{\infty }\\ t<2\end{array}}{lim}\frac{2e^{(t-2)x}}{t-2}-\left(\frac{2e^{(t-2)(0)}}{t-2}\right)$
• $M_x(t)=\underset{\begin{array}{c}x\to \mathrm{\infty }\\ t<2\end{array}}{lim}\underset{\to 0}{\underbrace{\frac{2e^{(t-2)x}}{t-2}}}-\left(\frac{2e^{(t-2)(0)}}{t-2}\right)$
• $M_x(t)=\frac{2}{2-t}$ for $t<2$
• Mean
  • $M_x^{\prime }(0)=\frac{d}{dt}{M}_x(t)=\frac{d}{dt}2(2-t{)}^{-1}=2(2-t{)}^{-2}$
  • $\mu =E(x)=M_x^{\prime }(t)|{}_{t=0}$
  • $\mu =E(x)=\frac{2}{(2-t{)}^2}|{}_{t=0}$
  • $\mu =E(x)=\frac{2}{(2{)}^2}$
  • $\mu =E(x)=\frac{1}{2}$
• Variance
  • #tk
  • $M_x^{″}(0)$
  • $var(x)={\sigma }^2=E(x^2)-{\mu }^2$