Let $\theta_1,\dots,\theta_N$ be a collection of independent RVs, ditributed uniformly on $[0,1]$. Further let $\theta^{(r)}$ be the $r$th order statistic where $\theta^{(1)}\leq\dots \theta^{(r)}\leq\dots \theta^{(N)}$.

Is the following reasoning correct?

$\mathbb{E}[\theta_i \vert \theta_i\geq \theta^{(m)}] = \mathbb{P}(\theta_i =\theta^{(m)})\mathbb{E}[\theta_i \vert \theta_i=\theta^{(m)}] + \dots +\mathbb{P}(\theta_i =\theta^{(N)})\mathbb{E}[\theta_i \vert \theta_i=\theta^{(N)}]\\ =\frac{1}{N}\sum_{l=m}^{N}\mathbb{E}[\theta^{(l)}] =\frac{1}{N}\sum_{l=m}^{N}\frac{l}{N+1}$

Thanks in advance!


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