I am studying 'change of variable for random vairables' reading your lecture 4 slides, The University of Leicester:EC2019 Sampling and Inference.
I could understand 'the change of variables technique' for random variables thanks to the slides which is much more detailed than others. I cannot, however, understand an formula in the bottom of page 3. Why does the absolute value symbol disappear?
Expectation of a function of random variable. Let y=y(x) be a function of x∼f(x). The value of E(y) can be found without first determining the p.d.f g(y) of y. Quite simply, there is $E(y)= \int_{x} y(x) f(x) dx $. If y=y(x) is a monotonic transformation of x, then it follows that $E(y)= \int_{y} yg(y) dy = \int_{y} y f\{x(y)\} | \frac{dx}{dy}| dy = \int_{x} y(x)f(x)dx $ which establishes a special case of the result.
