Question

概率与数理统计，第2题

Answer 1

(1)
F(x)
=A.e^x ; x<0
=B ; 0≤x<1
=1-Ae^(-(x-1)) ; x≥1
f(x) =F'(x)
=A.e^x ; x<0
=0 ; 0≤x<1
=Ae^(-(x-1)) ; x≥1
∫(-∞-> +∞) f(x) = 1
∫(-∞-> 0) A.e^x dx +∫(1-> +∞) Ae^(-(x-1)) dx =1
A - A [ e^(-(x-1)) ]|(1-> +∞) =1
2A =1
A=1/2
F(1-) = B
F(1+) = lim(x->1) (1/2)e^(-(x-1)) = 1/2
=> B=1/2
ie
A=B =1/2
(2)
P(X>1/3)
=1-F(1/3)
=1 - 1/2
=1/2
(3)
f(x) =F'(x)
=(1/2).e^x ; x<0
=0 ; 0≤x<1
=(1/2)e^(-(x-1)) ; x≥1