[m]F(x)= ∫ ^{x}_{- ∞ }f(x)dx[/m]
[b]При x ≤0[/b]
[m]F(x)= ∫ ^{x}_{- ∞ }0dx=0[/m]
[b]При 0 < x ≤π[/b]
[m]F(x)= ∫ ^{x}_{- ∞ }f(x)dx=∫ ^{0}_{- ∞ }0dx+∫ ^{x}_{0 }\frac{sinx}{2}dx=\frac{1}{2}(-cosx)| ^{x}_{0 }=\frac{1}{2}(-cosx+1)[/m]
[b]При x >π [/b]
[m]F(x)= ∫ ^{x}_{- ∞ }f(x)dx=∫ ^{0}_{- ∞ }0dx+∫ ^{π}_{0 }\frac{sinx}{2}dx+∫ ^{x}_{π}dx=\frac{1}{2}(-cosx)| ^{π}_{0 }) =\frac{1}{2}(-cos(-π)+cos0)=1[/m]
Получаем:
а)
[m]F(x)\left\{\begin {matrix}0, x ≤0\\\frac{1-cosx}{2}, 0 < x ≤π\\1, x>π \end {matrix}\right.[/m]
б)
[m]P(0 < X < \frac{π}{4})=F(\frac{π}{4})-F(0)=\frac{1-cos\frac{π}{4}}{2}-0=\frac{2-\sqrt{2}}{4}[/m]