The probability density function ("PDF") describes the probability law of a continuous [[Random Variable]]. Unlike a [[Probability Mass Function]], the PDF does not give the probability that the r.v. $X$ takes on a specific value $x$ (since the probability of any single point is zero). Instead, it specifies the *probability density*, which is the *probability per unit length* at $x$. To compute a probability, we calculate the area under the "curve" over a range $[a,b]$. We need an integral for that. $ \mathbf P(a \le x \le b) = \int_a^b f_X(x) \, dx $ ![[probability-density.png|center|400]] Probabilities of small intervals can be approximated by calculating the area of a rectangle: ![[probability-density-small-interval.png|center|300]] The [[Probability Axioms#Axioms|probability axioms]] for single probabilities have to hold for a PDF as well: * *Non negativity:* $f_X(x) \ge 0$ * *Normalization:* $ \int_{-\infty}^\infty f_X(x) \, dx=1 $ **Key Properties:** 1. Probabilities of single points are always $0$. $ \begin{align} \mathbf P(b \le X \le b+\delta)&= f_X(x)*\delta \\ \mathbf P(b \le X \le b+0)&= f_X(x)*0 \\ \mathbf P(b)&=0 \end{align} $ 2. It does not matter if a range $[a,b]$ includes or excludes the thresholds. $ \begin{align} \mathbf P(a \le X \le b)&= \overbrace{\mathbf P(X=a)}^{0}+ \overbrace{\mathbf P(X=b)}^{0}+ \mathbf P(a<X<b) \\ \mathbf P(a \le X \le b) &= \mathbf P(a<X<b) \end{align} $