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?
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.
