Cumulative Distribution Functions

The cumulative distribution function is defined as the probability that a random variable \(X\) is less than \(x\), i.e.,

\[F_X(x)=\mathbf{P}(X\leq x)=\int_{-\infty}^{x}f_X(t)\,\mathrm{d}t.\]

Using the fundamental theorem of calculus, we can calculate the probability density function from the cumulative distribution function, i.e.,

\[f_X(x)=\dfrac{\mathrm{d}F_X(x)}{\mathrm{d}x}.\]

Properties of the CDF

• It is non-decreasing, i.e., if \(y\geq x\), then \(F_X(y)\geq F_X(x)\).
• As \(x\rightarrow -\infty\), \(F_X(x)\rightarrow 0\).
• As \(x\rightarrow +\infty\), \(F_X(x)\rightarrow 1\).