normal distribution prove OR disprove E[x]=μ
i believe the claim is true but i'm not sure how to go along to prove it...
The pdf of a normal distribution is:
to calculate the mean, you can evaluate by making a one to one substitution ..
The Right hand side of the above equation is zero..since it is the expected value of a standard normal dist. And the proof, like in most Mathematical Statistics books, should be in your book as an example.
it can be proved that this is equal to zero by using the fact that the above is an odd function. the proof of this should be in the OPs book. If not, they should show their attempt on the problem.
and after finding the mean of , a standard random variable, you can use that fact (as done in post #2) to find the mean of another normal random variable X with parameters
The easiest way to prove it is to just take the anti derivative and evaluate.