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Math Help - Conditional expectation where I need to find E[X^2|Y=0.7]

  1. #1
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    Conditional expectation where I need to find E[X^2|Y=0.7]

    The function is x+y where both x and y are between 0 and 1.

    Am I correct in saying if E[X|Y] = f(x,y)/h(y)

    where h(y) is the marginal probability density of Y that:

    E[X^2|Y]= [f(x,y)]^2/h(y)

    Sorry if this is a stupid question but I don't want to try and solve it in different ways and there still be some ambiguity as to what the correct way to tackle the question is.

    Thanks.
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  2. #2
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    Re: Conditional expectation where I need to find E[X^2|Y=0.7]

    no i dont think so. As your wrote it, your E(X|Y) would almost certainly be a function of X, so you know that it is incorrect. E(X|Y) should be a function of Y only (or a constant).


    Lets start from the definition of expectation for any random variable, x: E(X) = \int x f_x(x) dx

    Similarly for the conditional variable "X|Y"

    E(X|Y) = \int x f_{x|y}(x) dx

    You can make this substitution f_{x|y}(x) = \frac{f_{xy}(x,y)}{f_y(y)}

    to get:
    E(X|Y) = \int x \frac{f_{xy}(x,y)}{f_y(y)} dx = \frac{1}{f_y(y)}\int x {f_{xy}(x,y) dx

    is this what you meant?

    E(X^2) is similar except the starting definition is E(X^2) = \int x^2 f(x) dx

    You try and finish. If you need further help, be sure to post the density functions you are supposed to be manipulating.
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  3. #3
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    Re: Conditional expectation where I need to find E[X^2|Y=0.7]

    Thanks for the reply springfan.

    If it was E[x|y] I would find the marginal density of y and then take the orginal function and put it over the marginal density of y right?

    I'm still a little perplexed as to how to do it for E[X^2|Y].

    So for f(x,y)= x+y if I needed to get the conditional expectation E[X|Y] it would be (x+y)/(xy)+(y^2/2)? So how does that change when it becomes E[X^2|y].
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  4. #4
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    Re: Conditional expectation where I need to find E[X^2|Y=0.7]

    your posts are hard to follow because you:
    1) havn't told me the densities of X and Y, or their joint density
    2) dont put any integral signs in what you write.

    E[X|Y] = (x+y)/(xy)+(y^2/2)
    i dont see where this could have come from, i dont think its right. If this is supposed to be the final answer, it should not be a function of X. If you think otherwise, you are fundamentally misunderstanding what an expectation is and you may want to review your notes.


    The integral you need to do is:
    E(X^2|Y) = \int x^2 f_{x|y}(x,y) dx

    this can be re-written as (same method as post #2)

    E(X^2|Y) = \frac{1}{f_y(y)}\int x^2 f_{x,y}(x,y) dx



    IF you need further help then post the the density functions of X and Y, or the joint density function
    Last edited by SpringFan25; December 13th 2011 at 01:28 PM.
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  5. #5
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    Re: Conditional expectation where I need to find E[X^2|Y=0.7]

    The function is x+y where both x and y are between 0 and 1.
    Just noticed this in post #1. if "the function" means "the joint density function" then:

    step 1: find the marginal density of y

    Spoiler:

    f_y = \int_0^1 (x+y) dy = y + 0.5


    step 2: find the conditional density x|y
    Spoiler:

    f_{x|y} = \frac{f_{x,y}(x,y)}{f_y(y)}  =\frac{x+y}{y+0.5}


    step 3: evaluate the integral:

    E(X^2|Y) = \int_0^1 x^2 f_{x|y} dx

    Spoiler:

    Hint: you will get a fraction inside the integral. The denominator doesn't depend on X so can be taken outside the integral
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  6. #6
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    Re: Conditional expectation where I need to find E[X^2|Y=0.7]

    Apologies for being so vague, here is the question I need to solve, perhaps this would make it clearer.



    EDIT: I have re-read my notes and I think I understand how to do this now, thank you for your help and the solution.
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  7. #7
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    Re: Conditional expectation where I need to find E[X^2|Y=0.7]

    sorry, i saw the the density you put in post #1 while you were typing that. Try and follow the instructions in post #5.
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  8. #8
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    Re: Conditional expectation where I need to find E[X^2|Y=0.7]

    Quote Originally Posted by SpringFan25 View Post
    sorry, i saw the the density you put in post #1 while you were typing that. Try and follow the instructions in post #5.
    I think I have solved this question correctly, thank you for your help.

    Whilst this thread is new, could you give me some advice on something else please?



    I know I need to find the expectations of X1, X2 and X3 to find the answers first off, to do this do I simply integrate x*f(x1,x2,x3) w.r.t x1 then x2 then x3?
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