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Let $X$ ~ Uniform(0,1) with pdf $f(x)$:

The joint pdf of the 1st and $n$th order statistics is, say, $g(x_\left(1\right), x_\left(n\right))$:

where I am using the OrderStat function from the mathStatica package for Mathematica to automate the nitty gritties for me (I am one of the authors of the former).

... and with domain of support:

We seek $P(X_\left(n\right)- X_\left(1\right) > d)$:

All done.

Let $X$ ~ Uniform(0,1) with pdf $f(x)$:

The joint pdf of the 1st and $n$th order statistics is, say, $g(x_\left(1\right), x_\left(n\right))$:

where I am using the OrderStat function from the mathStatica package for Mathematica to automate the nitty gritties for me (I am one of the authors of the former).

... and with domain of support:

We seek $P(X_\left(n\right)- X_\left(1\right) > d)$:

All done.

Let $X$ ~ Uniform(0,1) with pdf $f(x)$:

The joint pdf of the 1st and $n$th order statistics is, say, $g(x_\left(1\right), x_\left(n\right))$:

where I am using the OrderStat function from the mathStatica package for Mathematica to automate the nitty gritties for me (I am one of the authors of the former).

... and with domain of support:

We seek $P(X_\left(n\right)- X_\left(1\right) > d)$:

All done.

Just as a side note, the solution obtained here is different to that of Carlo's posting:

whereas I obtain, for the same values, $\frac{8}{9}$ [checked via Monte Carlo ]

Let $X$ ~ Uniform(0,1) with pdf $f(x)$:

The joint pdf of the 1st and $n$th order statistics is, say, $g(x_\left(1\right), x_\left(n\right))$:

where I am using the OrderStat function from the mathStatica package for Mathematica to automate the nitty gritties for me (I am one of the authors of the former).

... and with domain of support:

We seek $P(X_\left(n\right)- X_\left(1\right) > d)$:

All done.

Let $X$ ~ Uniform(0,1) with pdf $f(x)$:

The joint pdf of the 1st and $n$th order statistics is, say, $g(x_\left(1\right), x_\left(n\right))$:

where I am using the OrderStat function from the mathStatica package for Mathematica to automate the nitty gritties for me (I am one of the authors of the former).

... and with domain of support:

We seek $P(X_\left(n\right)- X_\left(1\right) > d)$:

All done.

Just as a side note, the solution obtained here is different to that of Carlo's posting:

whereas I obtain, for the same parameter values $\frac{8}{9}$ [Checked via Monte Carlo ]

Let $X$ ~ Uniform(0,1) with pdf $f(x)$:

The joint pdf of the 1st and $n$th order statistics is, say, $g(x_\left(1\right), x_\left(n\right))$:

where I am using the OrderStat function from the mathStatica package for Mathematica to automate the nitty gritties for me (I am one of the authors of the former).

... and with domain of support:

We seek $P(X_\left(n\right)- X_\left(1\right) > d)$:

All done.

Just as a side note, the solution obtained here is different to that of Carlo's posting:

whereas I obtain, for the same values, $\frac{8}{9}$ [checked via Monte Carlo ]