Piecewise Densities and Moments

HL only. A density is non-negative and its total area is one. Integrate each piece over its own interval; density height is not the probability of a single value.

Example: \(\mathrm{f}(x)=\begin{cases}kx&0\le x\le1,\\k(2-x)&1<x\le2,\\0&\text{otherwise}.\end{cases}\)

\(\begin{aligned}1&=\int_0^1 kx\,\mathrm{d}x+\int_1^2 k(2-x)\,\mathrm{d}x\\&=\dfrac{k}{2}+\dfrac{k}{2}=k\end{aligned}\)

With \(k=1\), \(P(X<0.5)=\int_0^{0.5}x\,\mathrm{d}x=\dfrac18\). Symmetry gives mean and median \(1\); the density is greatest at \(1\), so this is also the mode.

\(E(X^2)=\int_0^1x^3\,\mathrm{d}x+\int_1^2x^2(2-x)\,\mathrm{d}x=\dfrac76\).

\(\operatorname{Var}(X)=\dfrac76-1^2=\dfrac16\), and \(\operatorname{SD}(X)=\sqrt{\dfrac16}\). For \(Y=3X-2\), \(E(Y)=1\) and \(\operatorname{Var}(Y)=\dfrac32\).