= Probability density function
{title2=$f_X(x)$}
{wiki}
= Statistical probability density
{synonym}
A <probability density function> represents probabilities as areas under a nonnegative <measurable function>. For a real <random variable> with an <absolutely continuous probability distribution>, it satisfies
$$
\mathbb P(X\in A)=\int_A f_X(x)\,dx,\qquad
f_X(x)\geq0,\qquad \int_{\mathbb R}f_X(x)\,dx=1
$$
for every <Borel set> $A$. Its height is probability per unit of the outcome variable, not the probability of that exact outcome. Thus a <probability density function> may exceed one, while the probability of a <singleton> is zero. NIST describes the interval-probability interpretation through integration. https://itl.nist.gov/div898/handbook/eda/section3/eda362.htm[NIST: probability density functions and interval probabilities]{ref}.
For the example $f_Y(y)=2y$ on $[0,1]$, $f_Y(0.8)=1.6$, but $\mathbb P(Y=0.8)=0$ and $\mathbb P(0.7<Y<0.9)=0.9^2-0.7^2=0.32$. The <cumulative distribution function> is $F_Y(y)=y^2$ within that <interval>, connecting density height to the local accumulation rate.
Changing the scale changes the density height. If $Z=100Y$, then $f_Z(z)=f_Y(z/100)/100=z/5000$ on $[0,100]$. At the corresponding value $z=80$, its height is $0.016$, while $\mathbb P(70<Z<90)=0.32$ is unchanged. The factor $1/100$ compensates for the wider integration variable. This is a <change of variables>, not a change in the underlying probabilities.
A <probability density function> is determined only <almost everywhere>: changing its values on a <Lebesgue measure> zero set changes no probabilities. The <Cantor distribution> has no such representation, and a <mixed probability distribution> can have both a point mass and a density component, so differentiating its <cumulative distribution function> does not recover all its mass.
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