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Moment generating function of a (normally distributed) random variable

Discussion in 'Education' started by ozarka, Oct 8, 2018.

  1. ozarka

    ozarka Guest

    I am given 3 things:

    1. $Z$ follows a normal distribution $N(0,1)$
    2. $Y=e^{X}$
    3. $X=3-2Z$

    What is the moment generation function of $X$ and the $r^{th}$ moment of $Y$ ($E[Y^{r}]$)?

    My attempt:

    I know that $M_{X}(t)=E[e^{tX}]=E[e^{t(\mu+\sigma Z)}]=e^{\mu t + (\sigma ^2 t^2)/2}$. So by $X=3-2Z$, $3$ is $\mu$ and $-2$ is $\sigma$. Therefore, $M_X(t)=e^{3t+2t^2}$. And since $E[Y^{r}]=E[e^{rX}]=M_X(r)$, $E[Y^{r}]= e^{3r+2r^2}$?

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