4.1.3—PDF and CDF

For a continuous variable, the probability density function f(x) satisfies P(a≤X≤b)=∫ₐᵇf(x)dx, while the cumulative distribution function F(x)=P(X≤x)=∫_{−∞}ˣf(t)dt.

A density value is not itself a probability; probabilities are areas. The CDF is non-decreasing, approaches 0 and 1 at the ends, and where differentiable F′(x)=f(x).

If f(x)=2x on 0≤x≤1, then P(X≤0.5)=∫₀^.5 2x dx=0.25 and F(0.5)=0.25.

For a continuous variable P(X=x)=0, even though f(x) may be positive; use an interval or tail probability.