# Expectation of an exponential function

Why is the expectation of an exponential function: $$\mathbb{E}[\exp(A x)] = \exp((1/2) A^2)\,?$$

I am struggling to find references that shows this, can anyone help me please?

If anyone could enlighten me it would be great!

• Expectation with respect to what probability density function? Dec 14, 2012 at 16:31
• What is $A$ and $x$? Dec 14, 2012 at 16:34
• A is a constant and x is a random variable that is assumed Gaussian Dec 14, 2012 at 16:39
• Yes: Expectation with respect to what probability density function Dec 14, 2012 at 16:40
• What is $E[x]$ and $\mathrm{Var}(x)$? Dec 14, 2012 at 16:46

Let $X\sim\mathcal{N}(0,1)$ and $a\in\mathbb R$. Then \begin{align*} E[\exp(aX)]&=\int_{\mathbb R}\frac{1}{\sqrt{2\pi}}\exp\left(-\frac{1}{2}x^2\right)\exp(ax)\,\mathrm dx=\int_{\mathbb R}\frac{1}{\sqrt{2\pi}}\exp\left(-\frac{1}{2}(x-a)^2+\frac{1}{2}a^2\right)\\ &=\exp\left(\frac{1}{2}a^2\right)\int_{\mathbb{R}}\frac{1}{\sqrt{2\pi}}\exp\left(-\frac{1}{2}(x-a)^2\right)=\exp\left(\frac{1}{2}a^2\right) \end{align*} because $$x\mapsto \frac{1}{\sqrt{2\pi}}\exp\left(-\frac{1}{2}(x-a)^2\right)$$ is the density of an $\mathcal{N}(a,1)$ distribution.