Probability density or distribution function

A continuous random variable is a theoretical representation of a continuous variable such as height, mass or time.

A continuous random variable is specified by its probability density function (p.d.f.) written as \(\mathrm{f}(x)\).

If \(X\) is a continuous random variable with probability density function \(\mathrm{f}(x)\) valid over the range \(a \le x \le b\), then

1. \(\mathrm{f}(x) \ge 0\)
2. \(\int_{\text{all}\hspace{0.5em} x} \mathrm{f}(x) \, \mathrm{d}x = 1\), i.e,
   \[ \int_{a}^{b} \mathrm{f}(x) \, \mathrm{d}x = 1. \]

3. \(\mathrm{P}(X \le x_1) = \int_{a}^{x_1} \mathrm{f}(x) \, \mathrm{d}x\), where \(a \le x_1 \le x_2 \le b\).
4. \(\mathrm{P}(X \ge x_2) = \int_{x_2}^{b} \mathrm{f}(x) \, \mathrm{d}x\), where \(a \le x_1 \le x_2 \le b\).
5. \(\mathrm{P}(x_1 \le X \le x_2) = \int_{x_1}^{x_2} \mathrm{f}(x) \, \mathrm{d}x\), where \(a \le x_1 \le x_2 \le b\).

Note:

\[ \begin{align*} \mathrm{P}(x_1 \le X \le x_2) & = \mathrm{P}(X = x_1) + \mathrm{P}(x_1 < X < x_2) + \mathrm{P}(X = x_2) \\ & = 0 + \mathrm{P}(x_1 < X < x_2) + 0 \\ & = \mathrm{P}(x_1 < X < x_2). \end{align*} \]

\[\therefore \mathrm{P}(x_1 \le X \le x_2) = \mathrm{P}(x_1 < X < x_2) = \mathrm{P}(x_1 \le X < x_2) = \mathrm{P}(x_1 < X \le x_2).\]