Distribuzione di probabilità trapezioidale.

Figure: Distribuzione trapezioidale.

La probabilità è data da:

$$\left\{ \begin{array}{l} p\left(x\right)=-\frac{4\left(x-a_{-}\ri... ...ad\forall x\in\left[\frac{a_{-}+a_{+}}{2}+a\beta,a_{+}\right]\end{array}\right.$$

dalla quale si ha:

$$\int_{-\infty}^{+\infty}p\left(x\right)dx=\int_{-\infty}^{a_{-}}p... ...x+\int_{a_{-}}^{a_{+}}p\left(x\right)dx+\int_{a_{+}}^{+\infty}p\left(x\right)dx$$

ed ancora, poiché è:

$$\int_{-\infty}^{+\infty}p\left(x\right)dx=\int_{a_{+}}^{+\infty}p\left(x\right)dx=0$$

si ha:

$$\int_{-\infty}^{+\infty}p\left(x\right)dx=\int_{a_{-}}^{a_{+}}p\l... ...}p\left(x\right)dx+\int_{\frac{a_{-}+a_{p}}{2}+a\beta}^{a_{+}}p\left(x\right)dx$$

sostituendo si ha:

$$\int_{a_{-}}^{\frac{a_{-}+a_{p}}{2}-a\beta}p\left(x\right)dx=\int... ...{-}\right)^{2}}dx=\frac{a_{+}-a_{-}-2a\beta}{2\left(a_{+}-a_{-}+2a\beta\right)}$$

ed ancora:

$$\int_{\frac{a_{-}+a_{p}}{2}-a\beta}^{\frac{a_{-}+a_{p}}{2}+a\beta... ...frac{2}{\left(a_{+}-a_{-}\right)+2a\beta}dx=\frac{4a\beta}{a_{+}-a_{-}+2a\beta}$$

ed infine:

$$\int_{\frac{a_{-}+a_{p}}{2}+a\beta}^{a_{+}}p\left(x\right)dx=\int... ...{-}\right)^{2}}dx=\frac{a_{+}-a_{m}-2a\beta}{2\left(a_{+}-a_{-}+2a\beta\right)}$$

per cui:

$$\int_{-\infty}^{+\infty}p\left(x\right)dx=\frac{a_{+}-a_{-}-2a\be... ...+}-a_{-}+2a\beta}+\frac{a_{+}-a_{-}-2a\beta}{2\left(a_{+}-a_{-}2a\beta\right)}=$$

$$=\frac{a_{+}-a_{-}-2a\beta}{a_{+}-a_{-}+2a\beta}+\frac{4a\beta}{a_{+}-a_{-}+2a\beta}=\frac{a_{+}-a_{-}-2a\beta+4a\beta}{a_{+}-a_{-}+2a\beta}=1$$

mentre:

$$u^{2}\left(x\right)=\frac{a^{2}\left(1+\beta\right)}{6}$$