Probability Density Functions

A continuous random variable \(X\) is described by a probability density function \(f_X(x)\). In this case, however, the probability that it takes a single value, say \(a\), is zero, i.e.,

\[\mathbf{P}(X=a)=0.\]

Instead, we can take the probability that it is within a certain range, say between \(a\) and \(b\). This is defined by

\[\mathbf{P}(a\leq X\leq b)=\int_{a}^{b}f_X(x)\,\mathrm{d}x.\]

Non-negativity and Normalization

Like in the discrete case, the probability density function has to be non-negative and normalized, i.e.,

\[f_X(x)\geq 0\qquad\text{and}\qquad\int_{-\infty}^{\infty}f_X(x)\,\mathrm{d}x=1.\]