Question

两道概率论问题，请教大神，求详细过程，谢谢!

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Answer 1

f(x)= ∫ [0,x] e^(-x)dy = xe^(-x), x>0.
f(y|x)=f(x,y)/f(x) = e^(-x)/(xe^(-x)) = 1/x, 0<y<x.
现在忙. 先做这些.

追问

能不能写一下第（2）问的详细过程呢?

追答

见图1. f(x)=∫[0,x] e^(-x)dy = xe^(-x), x>0.

f(y|x)=f(x,y)/f(x) = e^(-x)/(xe^(-x)) = 1/x, 0<y<x.

F(x) =∫[0,x] f(a) da =∫[0, x] ae^(-a)da = 1-[e^(-x)](1+x),

Fx(1) = 1-2e^(-1).

 

见图2. f(y) =∫f(x,y)dx =∫[y, ∞] e^(-x)dx = e^(-y), 0<y<∞.

F(y) =∫[0, y] f(a)da =∫[0, y] e^(-a)da = 1-e^(-y), y>0.

FY(1) = 1-e^(-1).

P(X≤1|Y≤1) = P(X≤1,Y≤1)/P(Y≤1) = P(X≤1)/P(Y≤1)

= {Fx(1)} / {FY(1)} = {1-2e^(-1)} / {1-e^(-1)} = (e-2)/(e-1) = 0.418

见图1. f(x)=∫[0,x] e^(-x)dy = xe^(-x), x>0.

f(y|x)=f(x,y)/f(x) = e^(-x)/(xe^(-x)) = 1/x, 0<y<x.

F(x) =∫[0,x] f(a) da =∫[0, x] ae^(-a)da = 1-[e^(-x)](1+x),

Fx(1) = 1-2e^(-1).

 

见图2. f(y) =∫f(x,y)dx =∫[y, ∞] e^(-x)dx = e^(-y), 0<y<∞.

F(y) =∫[0, y] f(a)da =∫[0, y] e^(-a)da = 1-e^(-y), y>0.

FY(1) = 1-e^(-1).

P(X≤1|Y≤1) = P(X≤1,Y≤1)/P(Y≤1) = P(X≤1)/P(Y≤1)

= {Fx(1)} / {FY(1)} = {1-2e^(-1)} / {1-e^(-1)} = (e-2)/(e-1) = 0.418

追问

可以再写一下另一道题吗?

追答

此图还是第一题的. 第二题不熟. 对不起. 请别人吧.