Conditional expectation question

[busstud123]

Hey,

I would really appreciate your help with the following question:

X and Y each has a uniform distribution on [-u,u] and the two are independent.

could you please show how to compute E(X|X<Y)?

Thank you very much for your help!

Here's how far I got:

Pr(X=u|X<Y) = Pr(X=u,X<Y) / Pr(X<Y) = Pr(X=u,u<Y) / Pr(u<Y) = (since X and Y are independent) Pr(X=u)*Pr(Y>u) / Pr(Y>u) = Pr(X=u)

That doesn't seem to make intuitive sense to me though.

[Martingale]

Hey,

I would really appreciate your help with the following question:

X and Y each has a uniform distribution on [-u,u] and the two are independent.

could you please show how to compute E(X|X<Y)?

Thank you very much for your help!

Here's how far I got:

Pr(X=u|X<Y) = Pr(X=u,X<Y) / Pr(X<Y) = Pr(X=u,u<Y) / Pr(u<Y) = (since X and Y are independent) Pr(X=u)*Pr(Y>u) / Pr(Y>u) = Pr(X=u)

That doesn't seem to make intuitive sense to me though.

Are X,Y continuous uniform or discrete uniform?

[BGM]

I guess it should be continuous uniform on [-u, u]
Note ∀y∈(-u, u], ∀x∈[-u, y),
Pr{X ≤ x|X < y} = Pr{X ≤ x, X < y}/Pr{X < y} = Pr{X ≤ x}/Pr{X < y}
= [(x + u)/2u]/[(y + u)/2u] = (x + u)/(y + u)
fX|X<y(x) = dPr{X ≤ x|X < y}/dx = 1/(y + u)
*A simpler intuitive way is to note that X|X < y is just another uniform random variable
on [-u, y)
E[X|X < y] = ∫xdx/(y + u) = [y^2 - (-u)^2]/[2(y + u)] = (y - u)/2
As a result, y∈(-u, u]
E[X|X < Y] = ∫E[X|X < Y, Y = y]fY(y)dy = ∫[(y - u)/2][1/2u]dy
= (1/4u)[u^2/2 - u*u - (-u)^2/2 + u(-u) = -u/2

[busstud123]

Thank you very much. That's great! I see the approach now.

[Martingale]

I guess it should be continuous uniform on [-u, u]
Note ∀y∈(-u, u], ∀x∈[-u, y),
Pr{X ≤ x|X < y} = Pr{X ≤ x, X < y}/Pr{X < y} = Pr{X ≤ x}/Pr{X < y}
= [(x + u)/2u]/[(y + u)/2u] = (x + u)/(y + u)
fX|X<y(x) = dPr{X ≤ x|X < y}/dx = 1/(y + u)
*A simpler intuitive way is to note that X|X < y is just another uniform random variable
on [-u, y)
E[X|X < y] = ∫xdx/(y + u) = [y^2 - (-u)^2]/[2(y + u)] = (y - u)/2
As a result, y∈(-u, u]
E[X|X < Y] = ∫E[X|X < Y, Y = y]fY(y)dy = ∫[(y - u)/2][1/2u]dy
= (1/4u)[u^2/2 - u*u - (-u)^2/2 + u(-u) = -u/2

Hmmm.... I keep getting -u/3

[busstud123]

Would it be ok for you to show your steps?

[Dragan]

Hmmm.... I keep getting -u/3

Well, I also happen to keep getting -u/3. :)

[busstud123]

Thanks for posting guys. Is there any chance you could make your derivation publicly available? Or email me privately?

[Dragan]

Thanks for posting guys. Is there any chance you
could make your derivation publicly available?

Look here:

Let: U = min(X,Y), where min(X,Y)<=z.

Pr{min(X,Y) > z} = Pr{X>z} Intersect Pr{Y>z}.

Pr{U>z} = Pr{X>z}*Pr{Y>z}.

Pr{U>=z} = (1 - Fx(z))*(1 - Fy(z)).

Fu(z) = 1 - (1 - Fx(z))*(1 - Fy(z)).

Thus,

F_min(x,y) = Fx(z) + Fy(z) - Fx(z)*Fy(z).