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Let be $X \sim N(2,4)$. What is the mathematical expectation of $e^{X}$?

My approach:

Let be $Z=\exp(X)$, then

\begin{align*} \mathbb{F}_{Z}(z)&=\mathbb{P}(Z\leq z) \\&=\mathbb{P}(\exp(X)\leq z)\\&=\mathbb{P}(X\leq \ln z)\\ &=\mathbb{F}_X(\ln z) \end{align*}

So,

\begin{align} \mathbb{E}(Z)&=\int_{- \infty}^{\infty} z \cdot f_{Z}(z)\\&=\int_{- \infty}^{\infty} z \cdot \underbrace{f_{X}(\ln z)}_{\text{Is this correct?}}\\&=\frac{1}{\sqrt{2 \pi}}\cdot \int_{- \infty}^{\infty} z \cdot \exp\left ( - \frac{(\ln z - 2)^{2}}{8} \right ) dz \end{align}

My doubts:

  • Am I correct until this point?
  • If it is necessary to compute that integral, do you have any hint?
  • Is there another way, easier and faster, to compute that expectation?

I really appreciate your help! Thank you very much!

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    $\begingroup$ The integrand of "is this correct?" is not correct: a change of variables is slightly more complicated. The limits of the integral should also change since $\exp(X)$ is never negative. $Z$ has a log-normal distribution and one way to find its mean and other moments is to use the moment generating function of $X$ $\endgroup$
    – Henry
    Commented Sep 3, 2021 at 20:47
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    $\begingroup$ $\mathbb{E}[\exp({\text{Normal}(\mu,\sigma^2))}] = \mathbb{E}[\text{Lognormal}(\mu, \sigma^2)] = \exp\left(\mu+\frac{\sigma^2}{2}\right)$, see log-normal distribution. $\endgroup$ Commented Sep 3, 2021 at 20:56

1 Answer 1

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By definition of expectation,

$$\begin{align} \mathbb E[e^X] &= \int_{-\infty}^\infty e^x f_X(x)\,\mathrm dx \\[1ex] &= \frac1{4\sqrt{2\pi}} \int_{-\infty}^\infty e^{x-\frac1{32}(x-2)^2}\,\mathrm dx \\[1ex] &= \frac{e^{10}}{4\sqrt{2\pi}} \int_{-\infty}^\infty e^{-\frac1{32}(x-18)^2}\,\mathrm dx \end{align}$$

Can you see the PDF of another normal distribution here? (See LOTUS)

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    $\begingroup$ I think it's worth mentioning you are making use of the law of the unconscious statistician under the hood there. $\endgroup$ Commented Sep 3, 2021 at 21:24
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    $\begingroup$ Ha, I was not aware of this law! "... tendency to use the identity without realizing ..." is right. Added a link for reference $\endgroup$
    – user170231
    Commented Sep 3, 2021 at 21:52

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