Let $X,Y,X$ be random variables defined on the same probability space $(\Omega,F,P)$. Suppose that $X,Y,Z$ are independent and identically distributed and the common distribution is continuous. Prove that
$\displaystyle P\{X My try: well I've done this in a particular case, and as the problem says, it's true when
$X, Y, Z$ have exponential distributions with parameter $\lambda$. There it is easy because the problem reduces to calculating the integral
$\displaystyle \int_{0}^{\infty}\int_{x}^{\infty}\int_{y}^{\infty}\lambda^{3}e^{-\lambda(x+y+z)}dz dy dx.$ But I don't know how to do it in general. Any suggestions? Thanks beforehand.
Independent Random variables, identically distributed
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probability-theory
3 Answers
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Hint: ${\rm P}(X < Y < Z) = {\rm P}(X < Z < Y) = \ldots$.
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You should replace the integrand by the joint probability distribution $p_{XYZ}(x, y, z)$. Now $X$, $Y$ and $Z$ are independent, so
$p_{XYZ}(x, y, z) = p_X(x)p_Y(y)p_Z(z)$
besides they are identically distributed, so $p_X = p_Y = p_Z = p$. So I think you are back to almost the same calculations as you did in your particular case, the only property needed being
$\int_{-\infty}^{+\infty}p(x)dx=1$
Does that help?
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0The solution below is much, much more elegant. I'm ashamed! – 2011-05-11
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Symmetry rules the world, my friends... See this answer.