- #1

fluidistic

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## Homework Statement

Let X be a continuous random variable with parameters [itex]\langle x \rangle[/itex] and [itex]\sigma[/itex].

Calculate the probability density of the variable Y=exp(X). Calculate the mean and the variance of Y.

## Homework Equations

Reichl's 2nd edition book page 180:

[itex]P_Y(y)=\sum _{-\infty}^{\infty} \delta (y-H(x))P_X(x)dx[/itex] where H(x)=Y, so I guess, H(x)=exp (x).

## The Attempt at a Solution

I used the given formula and reached [itex]P_Y(y)=P_X (\ln y)[/itex]. Is this the correct answer?

I don't really know how to calculate the mean, [itex]\langle y \rangle[/itex]. I guess they want it in function of the mean of X, namely [itex]\langle x \rangle[/itex].

I know that [itex]\langle x \rangle=\int _{-\infty}^{\infty} xP_X(x)dx[/itex] and that [itex]\langle y \rangle=\int _{0}^{\infty} yP_Y(y)dy[/itex]. But this does not help me much. Any idea on how to continue further?