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What 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?

I am assuming Gaussian distribution.

A is a constant and x is a random variable that is gaussian distributed.

Damian
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  • What are you actually asking here? For help with a derivation? – jbowman Dec 14 '12 at 17:01
  • It's considered bad form to cross-post things to multiple SE sites (math.SE and here, in this case). Please chose one which you think is appropriate and post it there. This question has already been answered on math.SE. – Jonathan Christensen Dec 14 '12 at 17:52
  • Cross-posted at http://math.stackexchange.com/questions/258761/expectation-of-an-exponential-function. Although questions like this are (marginally) on topic here, this is a purely math question that has been asked and answered on that site, so I'm closing this copy of it. – whuber Dec 14 '12 at 17:58

1 Answers1

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Wikipedia's page on the log-normal distribution has the more general result for distributions with non-zero location parameter $\mu$.

It notes that, for the lognormal distribution defined as:

$$X = e^{\mu + \sigma Z}$$

with $Z$ a standard normal variable, the expectation is:

$$\mathbb{E}[X] = e^{\mu + \sigma^2/2}$$

Josh O'Brien
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