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The Rao-Blackwell theorem says that, for a sufficient statistic $T$ and a finite-second-moment estimator $\delta (X)$, the conditional estimator ${\delta }_T(T)=E_{\theta }[\delta (X)\mid T]$ can be chosen independently of the unknown parameter and has no greater mean squared error for estimating any specified function $g(\theta )$. Sufficiency makes the conditional distribution of the data given $T$ parameter-independent. Conditional expectation preserves the expectation, hence preserves unbiasedness. Writing $\delta -g=(\delta -{\delta }_T)+({\delta }_T-g)$, the cross term has expectation zero, because $E[\delta -{\delta }_T\mid T]=0$. ThereforeEquality holds exactly when $\delta ={\delta }_T$ almost surely. This proves the squared-loss form, including the usual unbiased-estimator variance statement.

Here $\theta >0$ is necessary for the stated uniform interval. Put $L=X_{(1)}={\min }_i{X}_i$ and $R=X_{(n)}={\max }_i{X}_i$. The joint density factors asThe factorization theorem for sufficient statistics shows that $T=(L,R)$ is sufficient. Also $E[X_1]=2\theta$, so $X_1/2$ is unbiased and has variance ${\theta }^2/12$.

For $n\ge 2$, conditional on distinct extrema $l,r$, exchangeability gives probability $1/n$ that $X_1$ is each endpoint; with probability $(n-2)/n$ it is one of the remaining observations, uniform on $(l,r)$. Its conditional mean is consequentlyThus the Rao-Blackwell estimator for a uniform scale interval iswhich is unbiased and has no greater mean squared error. For $n=1$, $L=R=X_1$ and the same formula reduces to the original estimator. As a quantitative check, the covariance of two uniform order statistics givesFor instance, after writing $X_i=\theta +2\theta U_i$, both extreme uniform variances are $n/[(n+1{)}^2(n+2)]$ and their covariance is $1/[(n+1{)}^2(n+2)]$. These formulas follow by integrating the joint extreme density $n(n-1)(r-l{)}^{n-2}$ on $0<l<r<1$; they also display strict improvement for $n>1$.

For $0\le u\le 1$, independence givesThus $U$ has probability density function $f_U(u)=2(1-u)$, and its expected value isPointwise, $U+V=X+Y$ and $UV=XY$. By linearity of expectation, $\mathbb{E}V=1-1/3=2/3$; by independence, $\mathbb{E}(UV)=\mathbb{E}X\mathbb{E}Y=1/4$. Consequently the covariance of two uniform order statistics isThe order statistics are dependent despite the independence of the original observations.

For independent and identically distributed random variables $U_1,\dots ,U_n$ with the continuous uniform distribution on $(0,1)$, the $r$th order statistic has Beta distributionIts probability density function, mean, and variance areThe probability density function is zero outside $[0,1]$; values assigned to its endpoints do not change any probability. Peter Kempthorne's MIT lecture notes derive the uniform order statistic law and these moments. ^[ref].

For $n=5$, the minimum, sample median, and sample maximum correspond to $r=1,3,5$. Their probability density functions are $5(1-t{)}^4$, $30t^2(1-t{)}^2$, and $5t^4$, respectively. The sample median therefore has Beta distribution $\mathrm{Beta}(3,3)$, mean $1/2$, and variance $1/28\approx 0.0357143$. Its cumulative distribution function isFor instance, $\mathbb{P}(U_{(3)}\le 1/4)=53/512\approx 0.103516$. The mean of the minimum is $1/6$, and the mean of the sample maximum is $5/6$. The fixed observed sample median $0.4$ in the order statistic example is one possible outcome; $1/2$ is the average over repeated statistical samples from this uniform distribution.

Each curve has area one. The extreme order statistics favor the ends, while the sample median favors the center. The endpoint density height $5$ is legitimate: the probability density function gives area per unit length, rather than point probability. In particular, a continuous uniform distribution has zero probability of producing an endpoint.

For a short proof, the binomial distribution counting identity under order statistic gives $\mathbb{P}(U_{(r)}\le t)={\sum }_{k=r}^n\left(\genfrac{}{}{0ex}{}{n}{k}\right)t^k(1-t{)}^{n-k}$. Taking its derivative cancels consecutive terms, leaving the displayed probability density function. Recognize its normalizing factorial coefficient as that of the Beta distribution, then substitute $\alpha =r$, $\beta =n+1-r$ into the Beta distribution mean and variance formulas. Although the original observations are independent, their order statistics generally are dependent; see the positive covariance of two uniform order statistics.