= Solution
The <cumulative distribution function> is
$$
\boxed{F_X(x)=\mathbb P(X\leq x)}.
$$
For an <absolutely continuous probability distribution>, the <probability density function> is a nonnegative integrable <function> $f_X$ with
$$
\boxed{F_X(x)=\int_{-\infty}^{x}f_X(u)\,du,\qquad\int_{-\infty}^{\infty}f_X(u)\,du=1}.
$$
Thus interval probabilities are <integrals> of $f_X$, and $F_X'=f_X$ almost everywhere, in particular at continuity points of $f_X$.
The word “continuous” in the source is understood in this density sense. A merely continuous <cumulative distribution function> need not possess a <probability density function>; it can instead describe a <singular continuous distribution>.
Back to article page
