I need some help on this:
E(X)=(1/(sigma(sqrt (2pi))) ∫(-infinity to infinity) xexp(-(x-m)^2/(2sigma^2)dx
Show E(X)=m.
I know to add and subtract m to the first x, but I don't know where to go from there.
Thanks for any help.
I need some help on this:
E(X)=(1/(sigma(sqrt (2pi))) ∫(-infinity to infinity) xexp(-(x-m)^2/(2sigma^2)dx
Show E(X)=m.
I know to add and subtract m to the first x, but I don't know where to go from there.
Thanks for any help.
Make the substitution $\displaystyle u = \frac{x-m}{\sigma \sqrt{2}}$.
After the appropriate substitutions and simplifying you should get:
$\displaystyle E(X) = \frac{1}{\sqrt{pi}} \int_{-\infty}^{+\infty} \sigma \, \sqrt{2} \, u \, e^{-u^2} + m e^{-u^2} \, du$.
The integral of the first bit is zero because $\displaystyle u \, e^{-u^2}$ is odd.
So $\displaystyle E(X) = \frac{m}{\sqrt{\pi}} \int_{-\infty}^{+\infty} e^{-u^2} \, du = m$.