## Can I do expectation like this

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!

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