Question

密度函数的题：设随机变量X的分布函数F(x)=A(1-e^-x),x>=0;F(x)=0,x<=0

求(1)常数A；（2）X的密度函数；（3）计算P(1<X<3)

Answer 1

1，F(+∞) = lim (x->+∞) A[1 - e^(-x)] = 1
A = 1
2，X的密度函数：f(x) = F'(x) = e^(-x) x >= 0
f(x) = 0 x < 0
3，P(1<x<3) = F(3) - F(1) = [1 - e^(-3)] - [1 - e^(-1)]
= e^(-1) - e^(-3) ≈ 0.318092

Answer 2

(1)令:F(+∝) = 1,得: A=1.
(2)求导数,得f(x)=F'(x)
x>=0时, f(x) =e^(-x) ,
x<0 时f(x) = 0.
(3) P(1<X<3) = F(3)-F(1) =[1-e^(-3)] - [1- e^(-1)] =e^(-1) -e^(-3)
或P(1<X<3)= 1/e - 1/e^3