Probability density function

A function $f(x)\ge0$ with $\int f(x)\,dx=1$ over its domain, whose integral over an interval gives a probability.

A probability density function (PDF) $f(x)$ describes a continuous random variable's distribution: $f(x)\ge0$ everywhere, $\int_{-\infty}^{\infty}f(x)\,dx=1$ (total probability is $1$ ), and $P(a\le X\le b)=\int_a^b f(x)\,dx$ — the PROBABILITY that $X$ falls in an interval is the AREA under $f$ over that interval, not $f$ 's value at any single point (which is always technically $0$ for a continuous variable).

Examples

For $f(x)=\frac{2x}{b^2}$ on $[0,b]$ : check $\int_0^b\frac{2x}{b^2}\,dx=1$ confirms it's a valid density.