# Expectation of a sequence of gaussian variables

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 $$x_k$$.

## 1 Answer

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

• This is exactly what I was looking for, thank you for the hints and clearing up my thinking. I appreciate it. – TheRefrigerator Apr 24 '19 at 21:30