(統計) 求累積機率密度函數

 $X，r.v.$

 $p.d.f.$

$$f(x)=\left\{ \begin{array}{l}\cfrac{2x}{21} &, 0\leq x\leq 3 \\ \cfrac{1}{2}-\cfrac{x}{14}  &,  3\leq x\leq 7 \\ 0 &, otherwise \end{array}\right.$$

求  c.d.f

(1).

$if $ $0\leq x\leq 3,$

$F(x)=0+\displaystyle\int_0^x\cfrac{2t}{21} dt=\left[\cfrac{t^2}{21}\right]^x_0=\cfrac{x^2}{21}$

$if $ $3\le x\leq 7,$

$F(x)=0+\displaystyle\int_0^3\cfrac{2t}{21} dt+\displaystyle\int_3^x\cfrac{1}{2}-\cfrac{t}{14} dt=\cfrac{-3}{4}+\cfrac{1}{2}x-\cfrac{1}{28}x^2$

$$c.d.f.\quad F(x)=\left\{ \begin{array}{l}0 &, x\lt0 \\ \cfrac{x^2}{21} &,  0\leq x\leq 3 \\  \cfrac{-3}{4}+\cfrac{1}{2}x-\cfrac{1}{28}x^2&, 3\le x\leq 7\\1 &, x\gt 7  \end{array}\right.$$