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All $X_1,X_2,X_3$ are independent and uniformly distributed on $[0,1]$. Find the probability of $X_2$ lying between $X_1$ and $X_3$

Is the following method correct? Find the $P(X_1<X_2<X_3) + P(X_3<X_2<X_1)$

Find $P(X_1<X_2<X_3)$

Fix $X_2$. Denote the point on $[0,1]$ as $x_2$. The probability of $P(X_1<X_2)$ is $x_2-0$ and the $P(X_2<X_3)=1-x_2$

$P(X_1<X_2<X_3)=(1-x_2)(x_2)$

GRS
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3 Answers3

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You can solve this problem by symmetry. The following events are distinct

$i$) $X_1$ lies between $X_2$ and $X_3$

$ii$) $X_2$ lies between $X_1$ and $X_3$

$iii$) $X_3$ lies between $X_1$ and $X_2$

and they are complete in the sense that one of them is always true, $P(i) + P(ii) + P(iii) =1$.

Because of the symmetry they are equivalent and we have $P(i) =P (ii)= P(iii)$ and thus

$$P(ii) = \frac13.$$

Fabian
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    This, plus the fact that ties have zero probability. – Did Jan 19 '16 at 12:48
  • Thanks @Did for the clarification! So in formulas, e.g., $P(X_1= X_2 < X_3) =0$. – Fabian Jan 19 '16 at 13:04
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    Just to make the answer more explicit: the uniform distribution is a red herring. It is only important that $X_1, X_2,X_3$ are i.i.d. distributed and that the ties have zero probability. If this is given, the probability for the event is always $1/3$. – Fabian Jan 19 '16 at 13:09
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Each of the three values $X_1, X_2, X_3$ has an equal chance of being the middle value.

paw88789
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You were close, but you just needed to integrate the variable over its support interval.

$$\begin{align}\mathsf P(X_1<X_2<X_3) & = \int_0^1 f_{X_2}(x_2)\; F_{X_1}(x_2)\; (1-F_{X_3}(x_2))\operatorname d x_2 \\[1ex] & = \int_0^1 1\cdot x_2\cdot(1-x_2)\operatorname d x_2 \\[1ex] & = {\left[\tfrac 1 2 {x_2}^2-\tfrac 1 3{x_2}^3 \right]}_{x_2=0}^{x_2=1} \\[1ex] & = \tfrac 1 6 \end{align}$$

And likewise $\mathsf P(X_3<X_2<X_1) =\tfrac 1 6$ so the probability that $X_2$ lies between $X_1$ and $X_3$ is $\tfrac 1 3$; as we should anticipate by symmetry -- we have three independent, identically distributed, continuous random variables , so any one of them will be between the other two with equal probability.

Graham Kemp
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