integrating product of PDF and CDF I am trying to show that the following integral:
$$
\int_{-\infty}^a F(x)~f(x)~dx = \frac{F(a)}{2!}
$$
Where $F$ is the cumulative distribution function of some continuous random variable X, and $f$ is the probability density function. Not quite sure how this conclusion was reached?

Clearly $F'(x)=f(x)$ (wherever CDF is continuous).
Substitute $F(x)=t$. Then $dt=f(x)dx$.
$$\int_{-\infty}^a F(x)~f(x)~dx \rightarrow \int_{0}^{F(a)} t~dt= \frac{(F(a))^2}{2}$$
