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Expected Value (joint pdf)

  1. Nov 30, 2013 #1
    1. The problem statement, all variables and given/known data
    A machine consists of 2 components whose lifetimes are X and Y and have joint pdf,
    [tex] f(x,y)=1/50[/tex] w/ [tex]0<x<10[/tex], [tex]0<y<10[/tex],[tex]0<x+y<10 [/tex]
    Calculate the expected value of [tex]X [/tex] given [tex]Y=5[/tex].

    2. Relevant equations
    [tex]E[X|Y]= \int_{-inf}^{inf} x f(x,y)/f(y) dx[/tex]


    3. The attempt at a solution
    [tex]f(y) = \int_{0}^{10-y} 1/50 dx = (1-y)/5[/tex]

    [tex]E[x|y] = \int_{0}^{10-y} x/(10-10y) dx [/tex]

    This is where I am confused. How do i set the limits on the integral where I actually compute the expectation? Is x=0 to x=10-y right? I was thinking I want to just integrate as normal and plug in y=5 after the integration.
     
  2. jcsd
  3. Nov 30, 2013 #2

    Ray Vickson

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    Careful:
    [tex]f(y) = \int_{0}^{10-y} 1/50 dx = \frac{10-y}{50} \neq (1-y)/5 \leftarrow \text{ what you wrote} [/tex]

    So, you should have a different ##f(x|y)## from what you wrote, which means that ##E(X|Y=y)## will be different from what you wrote. And, finally, YES, you should put y = 5 after the integration (or even before the integration, since y is kept constant throughout).
     
  4. Nov 30, 2013 #3
    thank you!

    OK, thanks for that. OK, just to be clear, can you tell me if this looks right?

    [ itex ]E[X|Y=5] = \int_{0}^{10} x E[X|Y=5] dx \int_{0}^{1} x * 1/5 dx = \left[ \frac{10}{2}x^2 \right]_{0}^{10} = 10[ /itex ]

    Not sure why I cant get this latex to work but I am getting 10 for my answer. Does that seem right?
     
  5. Nov 30, 2013 #4

    Ray Vickson

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    Edited version of the above:
    [tex]E[X|Y=5] = \int_{0}^{10} x E[X|Y=5] dx \int_{0}^{1} x * 1/5 dx = \left[ \frac{10}{2}x^2 \right]_{0}^{10} = 10[/tex]
    (obtained by replacing "[ itex ]" by "[tex ]" (remove the space between 'x' and ']' and replacing "[ /itex ]" by "[/tex ]"', again removing the final space. If you want in-line formulas (such as produced by "[itex ]"---no spaces) it is easier to use "# #" (no space between the two #s) at each end of the formula.

    Anyway, your formula is incomprehensible to me, and I cannot figure out why you would ever assume it is correct. It is essentially saying (for ##A = E[X|Y=5]##) that
    [tex] A = \int_0^{10} A x \, dx \cdot \int_0^1 \frac{x}{5} dx.[/tex]
    and that says
    [tex] A = A \frac{100}{2} \cdot \frac{1}{2} \frac{1}{5} = 50 A,[/tex]
    which could only be true for A = 0.

    Go back to square one, and go carefully. Just take your time, and check every step.
     
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