Question

Hello,

I am studying random variables.

Question is this: if rv X & a function g is known, what is the pdf of random variable Y = g(x)?

in the textbook answer is explained as follows.

P[y ≤ Y ≤ y + dy] = P[x ≤ X ≤ x + dx]

F_y(y + dy) - F_y(y) / dy dx = F_x(x + dx) - F_x(x) / dx dy

why is left side of dx & right side of dy exists in above equation?

Answer 1

What you're looking at is known as "the transformation theorem" and is just an integral change of variables written in probability notation.

Suppose g is an increasing function and Y = g(X). Then

F_Y(y) = P( g(X) < y ) = P( X < g^{-1}(y) ) = F_X( g^{-1}(y) )

To obtain the PDF, differentiate both sides of the equation above:

f_Y(y) = f_X( g^{-1}(y) ) D_y ( g^{-1}(y) )

where D_y means derivative with respect to y. Now if g were a decreasing function we'd have

F_Y(y) = P( g(X) < y ) = P( X > g^{-1}(y) ) = 1 - F_X( g^{-1}(y) )

and

f_Y(y) = f_X( g^{-1}(y) ) | D_y ( g^{-1}(y) ) |.

In the last line we would have -D_y. Since g is a decreasing function, it's derivative is negative and so the absolute values take care of the negative sign.