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Suppose we are given a sequence of iid Gaussian random variables $\{x_k \}$ with zero mean and unit variance. We create a new random variable $$X_n = e^{\sum_{n\ge k}x_k}.$$ How does one go about calculating the expectation $\mathbb{E}[X_n]$? I know that by definition that the expectation of a random variable $Y$ is $$\mathbb{E}[Y] = \int Y(\omega)\mathbb{P}(d\omega) $$ however, I am less clear on how to properly set up this integral in the case of the expectation of $X_n$, as it was created from $n$ random variables $x_k$.

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Hints:

  • You should be able to find the mean and variance of the sum $S$ of $n$ independent random variables with standard normal distributions; it too will have a normal distribution.

  • So you should be able to state the probability density function $f(s)$ for $S$

  • You want to find $$\mathbb E\left[e^S\right] = \int \limits_{-\infty}^{\infty}e^s f(s)\, ds$$ which should not be too difficult if you complete the square inside the exponential

  • There may be shortcuts, either using the moment generating function of a normal distribution or the mean of a lognormal distribution

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  • $\begingroup$ This is exactly what I was looking for, thank you for the hints and clearing up my thinking. I appreciate it. $\endgroup$ – TheRefrigerator Apr 24 at 21:30

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