Can I do expectation like this

[aegea]

Many thanks in advance

Suppose x is normal variable x~N(a,b)
and y=160*x^2
I need calculate E(y)=∫yf(y)d(y)
f(y) is the density function of y
how can I write it as an integral of x since we know x's distribution, I mean use the density function of x to substitute the original integral

Can I just use E(y)=E(160 x^2)= ∫ 160 x^2 f(x) d(x)
then I can use f(x), which is the density function of normal.

Seems right, but seems wrong, seems I use f(x) to substitute x directly!

 

[dyrich]

You can do it the way you suggested (it is logically correct), but I believe it will make more work for yourself (but I didn't look at it because there is a cleaner way).

Here are a few hints that should help:

Remember that the expected value function is a linear operator so

$$E(Y)=E(160X^2)=160E(X^2)$$

and

$$V(X)=E(X^2)-[E(X)]^2$$