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Thread: Is my proof correct?

  1. #1
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    Lightbulb Is my proof correct?

    I'm not sure if I proved the following correctly. (It is in a chapter on expectation, and I didn't use anything about expectation.) Any feedback would be greatly appreciated.

    Problem:
    Suppose that $\displaystyle X, Y,$ and $\displaystyle Z$ are nonnegative random variables such that $\displaystyle P(X+Y+Z \leq 1.3)=1$. Show that $\displaystyle X, Y,$ and $\displaystyle Z$ cannot possibly have a join distribution under which each of their marginal distributions is the uniform distribution on the interval $\displaystyle [0,1]$.


    Work:
    $\displaystyle \int_0^1\int_0^1 f(x,y,z)dxdy=z$
    $\displaystyle \int_0^1\int_0^1 f(x,y,z)dxdz=y$
    $\displaystyle \int_0^1\int_0^1 f(x,y,z)dydz=x$

    $\displaystyle \int_0^1 xdx=\frac{1}{2}$
    $\displaystyle \int_0^1 ydy=\frac{1}{2}$
    $\displaystyle \int_0^1 zdz=\frac{1}{2}$

    Answer:
    Integrating $\displaystyle f(x,y,z)$ over the cube gives $\displaystyle p=\frac{1}{2}$ instead of $\displaystyle 1$, therefore we have a contradiction. So $\displaystyle X, Y,$ and $\displaystyle Z$ cannot possibly have a join distribution under which each of their marginal distributions is the uniform distribution on the interval $\displaystyle [0,1]$.
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  2. #2
    MHF Contributor matheagle's Avatar
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    Re: Is my proof correct?

    I don't understand your integrals.

    IF you integrate out the other variables you obtain the marginal density.
    Hence

    $\displaystyle \int_0^1\int_0^1 f(x,y,z)dxdy=f_Z(z)=1$ IF you think that Z is U(0,1)
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  3. #3
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    Re: Is my proof correct?

    You're right. I had done $\displaystyle \int_0^z\int_0^1\int_0^1 f(x,y,z)dxdydz=F_Z(z)=z$ but wrote $\displaystyle \int_0^1\int_0^1 f(x,y,z)dxdy=z$. So $\displaystyle \int_0^1dz=1$.

    Now I'm at a loss on how to prove this.
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  4. #4
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    Re: Is my proof correct?

    $\displaystyle P(X+Y+Z \leq 1.3)=1 \Rightarrow E(X+Y+Z) \leq 1.3$
    If X, Y, and Z are $\displaystyle U(0,1)$ then $\displaystyle E(X+Y+Z)=E(X)+E(Y)+E(Z)=1.5$
    Contradiction. Therefore at least one of X, Y, or Z is not $\displaystyle U(0,1)$.
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