+ Reply to Thread
Results 1 to 2 of 2

Thread: Compuite P{X < Y} conditioned on the value of Y; intuition behind it.

  1. #1
    Points: 4,118, Level: 40
    Level completed: 84%, Points required for next Level: 32

    Posts
    12
    Thanks
    0
    Thanked 0 Times in 0 Posts

    Compuite P{X < Y} conditioned on the value of Y; intuition behind it.




    Dear people,

    I have a question concerning the following steps to compute P{X < Y}.
    The steps shown are the following:

    P{X < Y} = \int P(X < Y|Y = y)f_{Y}(y) dy
    = \int P(X < y|Y = y)f_{Y}(y) dy
    =\int P(X < y)f_{Y}(y) dy
    =\int F_{X}(y)f_{Y}(y) dy

    I do not completely get the intuition behind this.
    First you condition on Y = y, then you change the Y in y in the next step and then you remove the conditioning on Y = y because that obviously does not give you anymore information now that P(X < y). But couldn't you use P(X < y) right away?
    If someone could give some additional help that would be nice.

    Thank you!

  2. #2
    TS Contributor
    Points: 22,410, Level: 93
    Level completed: 6%, Points required for next Level: 940

    Posts
    3,020
    Thanks
    12
    Thanked 565 Times in 537 Posts

    Re: Compuite P{X < Y} conditioned on the value of Y; intuition behind it.


    I am not sure what exactly giving you trouble. I try to explain in the following way:

    The first step here is necessary - and is more general. I believe that in this question it was assumed that X, Y are independent so that the condition probability can be safely changed to the ordinary one in the 3rd line. But in general they can be dependent and you cannot do so.

    Or you may view this in other perspective:

    \Pr\{X < Y\} = \iint_{x<y} f_{X,Y}(x,y)d(x,y)

    = \int_{-\infty}^{+\infty} \int_{-\infty}^y f_{X|Y =y}(x|y)f_Y(y)dxdy

    = \int_{-\infty}^{+\infty} \left(\int_{-\infty}^y f_{X|Y =y}(x|y)dx\right) f_Y(y)dy

    = \int_{-\infty}^{+\infty} \Pr\{X < y|Y = y\}f_Y(y)dy

+ Reply to Thread

           




Posting Permissions

  • You may not post new threads
  • You may not post replies
  • You may not post attachments
  • You may not edit your posts






Advertise on Talk Stats