# Thread: how to tranform between functions in integral

1. ## how to tranform between functions in integral

Suppose x is normal variable x~N(a,b)
and y=160*x^2
I need calculate ∫f(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

Thank you!

2. Originally Posted by aegea
Suppose x is normal variable x~N(a,b)
and y=160*x^2
I need calculate ∫f(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

Thank you!
Read this: Noncentral chi-square distribution - Wikipedia, the free encyclopedia

3. Originally Posted by mr fantastic

Thank you. Is there a way I don't need to deal with chi-square dist since it's more difficult to get the integral.

Sorry for not saying it clear earlier. The original question was calculate E(y), since E(y)=∫yf(y)d(y)
can I just use E(y)=E(160x^2)=
∫160x^2f(x)d(x)
and f(x) is the density function of normal

Thank you

4. Originally Posted by aegea
Thank you. Is there a way I don't need to deal with chi-square dist since it's more difficult to get the integral.

Sorry for not saying it clear earlier. The original question was calculate E(y), since E(y)=∫yf(y)d(y)
can I just use E(y)=E(160x^2)=∫160x^2f(x)d(x)
and f(x) is the density function of normal

Thank you
Note that ∫x^2 f(x) dx = E(X^2).

Now note also that Var(X) = E(X^2) - [E(X)]^2 => E(X^2) = Var(X) + [E(X)]^2. And since you're told X ~ N(a,b) you know Var(X) and E(X) ....