Integration

[garcijon]

Hi All, I am having an issue with an integration which I should know how to solve. I am tasked with proving that the normal distribution's expectation is actually equal to it's mean.

This image described the task



I tried to use a change of variables where y = (x - μ). I tried integration by parts but somehow I can't arrive at the answer.

This is how I've setup the problem where y is defined as above

 

[HallsofIvy]

Hi All, I am having an issue with an integration which I should know how to solve. I am tasked with proving that the normal distribution's expectation is actually equal to it's mean.

This image described the task

View attachment 4036

I tried to use a change of variables where y = (x - μ). I tried integration by parts but somehow I can't arrive at the answer.

This is how I've setup the problem where y is defined as above

View attachment 4037

That's a good start! (Except for that leading constant- you have a "-" and no "\(\displaystyle \pi\)" under the square root.) Now, separate that as \(\displaystyle \frac{1}{\sigma\sqrt{2\pi}}\int_{-\infty}^{\infty}ye^{-\frac{y^2}{2\sigma^2}} ydy\)\(\displaystyle + \frac{\mu}{\sigma\sqrt{2\pi}} \int_{-\infty}^{\infty}\)\(\displaystyle e^{-\frac{y^2}{2\sigma^2}}dy\)

The first of those two integrals is easy- it's of an odd function from \(\displaystyle -\infty\) to \(\displaystyle \infty\). And you should recognize the second integral.