Normal distribution theoretical moments

#1
how we can show that the follwoing equility holds

E[(x-\mu)\sigma]=0

E[(x-\mu)^2\sigma^2-1]=0

E[(x-\mu)^3\sigma^3]=0

E[(x-\mu)^4\sigma^4-3]=0
 

Dason

Ambassador to the humans
#2
Hi! :welcome: We are glad that you posted here! This looks like a homework question though. Our homework help policy can be found here. We mainly just want to see what you have tried so far and that you have put some effort into the problem. I would also suggest checking out this thread for some guidelines on smart posting behavior that can help you get answers that are better much more quickly.
 
#3
hi thanks here is my work i have showed that equality hold for the first moment but i'm not sure for the rest
x~N(mu,sigma^2)
Z= x-mu/sigma ~ N(0,1)

E[Z]

we have pdf of normal distribution
1/(2*pi)^1/2 * exp(-1/2*x^2)

we have to show that

E(Z) ~ N(0,1)

integral (z*1/(2*pi)*1/2 *exp(-1/2*z^2) dx =

= -1/2*pi*1/2 *exp(-1/2*z^2) = 0

thanks
sorry i dont know how to type in lytex form
 

Dason

Ambassador to the humans
#4
Have you tried integration by parts? Use this along with the fact that the pdf of the normal distribution integrates to 1.