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In order to prove a theorem in my research, I would like to use a lemma on basic probability theory, but I don't know if it is correct.

For three random variables $X,Y$, and $Z$ not necessarily independent, does the equality $$ P(X \leq Y \leq Z) = P(X \leq Y)P(Y \leq Z)$$ hold?

$P(\cdot)$ is the probability operator.

Thanks in advance!

Alex Silva
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2 Answers2

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It would generally not be true even if they were independent.

For example if $X,Y,Z$ were identically and independently continuously distributed then

they can come in any order with equal probability so $ P(X \leq Y \leq Z) = \frac16$

but similarly $P(X \leq Y)P(Y \leq Z) = \frac12 \times \frac12 = \frac14$.

Henry
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    A lesson here is that even if $X$, $Y$, and $Z$ are independent, the events "$X\leq Y$" and "$Y\leq Z$" are not. – JiK Jul 30 '15 at 13:19
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Let $X=Z$, and $Y$ such that $P(X \le Y) = \frac{1}{2}$ and $P(X=Y)=0$, then in order for the equality to hold we must have \begin{eqnarray} 0 &=& P(X=Y) \\ &=&P(X \le Y \le Z) \\ &=& P(X \le Y) P(Y \le X)\\ &=& P(X \le Y) \left( 1- P(X<Y) \right) \\ &=& \frac{1}{4} \end{eqnarray} So the equality does not hold in general.

And that suchs $X$ and $Y$ exists, take $X=0$ and $Y$ the toss of a fair coin, where tails gets the value $1$ and heads the value $-3$.

Hetebrij
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