Cumulative Distribution Function of a CRV

If \(X\) is a continuous random variable with probability density function \(\mathrm{f}(x)\) defined for \(a \leq x \leq b\), then the cumulative distribution function (C.D.F) is given by \(F(t)\) where the interval is taken over all values of \(x \leq t\).

\[F(t) = \mathrm{P}(X \leq t) = \int_{a}^{t} \mathrm{f}(x) \, \mathrm{d}x\]

Note:

1. \[F(b) = \int_{a}^{b} \mathrm{f}(x) \, \mathrm{d}x = 1;\]
   \[F(a) = \int_{a}^{a} \mathrm{f}(x) \, \mathrm{d}x = 0.\]
2. \(\mathrm{P}(X \leq x_1) = F(x_1)\) , \(\mathrm{P}(X \leq x_2) = F(x_2)\) and \[\mathrm{P}(x_1 \leq X \leq x_2) = F(x_2) - F(x_1)\]