# Conditional expected value of order statistic

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?

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