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Let's say we have some continuous i.i.d random variables [itex]X_1, \ldots X_n[/itex] from a known distribution with some parameter [itex]\theta[/itex]

We then place them in ascending order [itex]X_{(1)}, \ldots X_{(n)}[/itex] such that [itex]X_{(i)}, < X_{(i+1)}[/itex].

We call this operation [itex]T(\mathbf{X})[/itex] where [itex]\mathbf{X}[/itex] is our vector [itex]X_1, \ldots X_n[/itex].

Now let's say we are interested in finding out whether [itex]P(\mathbf{X} = X| T(\mathbf{X}) = t)[/itex] where t and x are both vectors (and outcomes), depends on [itex]\theta[/itex].

By definition of conditional probability, we have:

[itex]P(\mathbf{X} = X| T(\mathbf{X}) = t) = \frac{P(\mathbf{X} = X, T(\mathbf{X}) = t)}{P(T(\mathbf{X}) = t)}[/itex]

Trying to find these 2 probabilities:

[itex]P(T(\mathbf{X}) = t)[/itex], this is the probability that my ascending ordering [itex]X_{(1)}, \ldots X_{(n)}[/itex] is a certain vector. This probability should simply be

[itex]\prod^{n}_{i=1}f(x_i)[/itex], since we already know that they are ordered.

[itex]P(\mathbf{X} = X, T(\mathbf{X}) = t)[/itex], this is the probability that my random vector [itex]\mathbf{X}[/itex] attains a certain (vector)value while the ordering attains a certain (vector)value. But this should also be equal to [itex]\prod^{n}_{i=1}f(x_i)[/itex].

So

[itex]P(\mathbf{X} = X| T(\mathbf{X}) = t) = \frac{P(\mathbf{X} = X, T(\mathbf{X}) = t)}{P(T(\mathbf{X}) = t)} = 1[/itex]. If this is correct, what does it mean that the probability is 1? Is it wrong? why?

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# Placing random variables in order

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