AHL 4.14 (HL)—Random variables and probability density

A probability density describes area, not height as probability.

For a continuous variable, P(a≤X≤b) is the area under f(x) between a and b; the total area is one and a single point has probability zero.

Example

If f(x)=2x on 0≤x≤1, P(X≤.5)=∫₀^.5 2x dx=.25.

Check the support and integrate over the interval before interpreting the result.

A density can exceed 1; only area, not height alone, is a probability.

Moment toolkit: for discrete $X$, $E(X)=\sum xP(X=x)$ and $E(X^2)=\sum x^2P(X=x)$; for continuous $X$, replace sums by integrals over its support. In both cases $Var(X)=E(X^2)-[E(X)]^2$ and $SD(X)=\sqrt{Var(X)}$. A continuous mode maximizes $f(x)$, while a median $m$ satisfies $\int_{-\infty}^{m}f(x)\,dx=1/2$. Under $Y=aX+b$, $E(Y)=aE(X)+b$ and $Var(Y)=a^2Var(X)$. For piecewise densities, integrate every relevant piece and require non-negative density with total area one; density height itself may exceed one.