Finding expectation and variance

[kensaurus]

So for example, if I have a random variable X, take it to be normally distributed.
How do you find the expectation and variance of the random variable e^X in terms of μ and σ?

Integrating the entire normal function with the f(x) is it?

 

[HallsofIvy]

Do you mean "mean" rather than "expectation"? In order to talk about "expectation", you have to say of what function of the random variable. The mean is the expectation of the function x- \int_{-\infty}^\infty xN(x,\mu,\sigma)dx. The variation is the expectation of the function (x- mean)^2, equal to \int_{-\infty}^\infty (x- mean)^2N(x,\mu,\sigma)dx.

 

[Ray Vickson]

So for example, if I have a random variable X, take it to be normally distributed.
How do you find the expectation and variance of the random variable e^X in terms of μ and σ?

Integrating the entire normal function with the f(x) is it?

Yes. However, it is best to find once and for all the expectation of exp(k*Z), where k is a constant and Z is a standard normal. Then, a general X has the form X = μ + σZ, so finding E[exp(X)] as exp(μ) * E[exp(σZ)] will be straightforward. As to Var(X): the easiest way is to use the standard theorem which that states that Var(Y) = E[Y²] - (EY)² and apply it to Y = exp(X) (with, of course, Y² = exp(2X)).

RGV