This gives us the new random variable \( y \) in the domain \( y \in [0,\infty) \) determined through the random variable \( x \in [0,1] \) generated by functions like \( ran0 \).

This means that if we can factor out \( \exp{(-y)} \) from an integrand we may have $$ \begin{equation*} I=\int_0^{\infty}F(y)dy=\int_0^{\infty}\exp{(-y)}G(y)dy \end{equation*} $$ which we rewrite as $$ \begin{equation*} \int_0^{\infty}\exp{(-y)}G(y)dy= \int_0^{1}G(y(x))dx\approx \frac{1}{N}\sum_{i=1}^NG(y(x_i)), \end{equation*} $$ where \( x_i \) is a random number in the interval \( [0,1] \).