Question

Найти функцию распределения F(х)

Answer 1

$F(x)= intlimits^{x}_{-infty } f(t), dt\\f(x)= left{begin{array}{l}0,; esli; ; ; x leq 1\x-frac{1}{2}; ,; esli; 1 textless x leq 2\0; ,; esli; ; ; x textgreater 2end{array}right\\a); ; x leq 1:; ; F(x)=intlimits^{x}_{-infty }, f(t), dt=int limits _{-infty }^{x}, 0, dt=0\\b); ; 1 textless x leq 2:; ; F(x)=intlimits^{x}_{-infty }, f(t)dt= intlimits^1_{-infty }, 0, dt+\\+intlimits^{x}_1, (t-frac{1}{2}), dt=0+(frac{t^2}{2}-frac{1}{2}, t)Big |_1^{x}=frac{x^2}{2}-frac{x}{2}=frac{1}{2}cdot (x^2-x)$

$c); ; x textgreater 2:; ; F(x)=int limits _{-infty }^1, 0, dt+intlimits^2_1, (t-frac{1}{2}), dt+intlimits^{x}_3, 0, dt=\\=(frac{t^2}{2}-frac{t}{2})Big |_1^2=frac{4}{2}-1-(frac{1}{2}-frac{1}{2})=2-1-0=1\\F(x)= left{begin{array}{l}0; ,; esli; xleq 1\frac{1}{2}cdot (x^2-x); ,; esli; 1<xleq 2\1; ,; esli; x>2end{array}right$