# Thread: Uniform distribution between (a,b): expected value of kth order statistic

1. ## Uniform distribution between (a,b): expected value of kth order statistic

Let $\displaystyle Y_1 ,...., Y_n$ be a random sample from an uniform distribution $\displaystyle U[a,b]$, both positives. Let $\displaystyle Y_{(k)}$ be the kth order statistic (that is the k-th smallest value extracted from that sample) from a sample of $\displaystyle n$ observations.

Find the expected value of $\displaystyle Y_{(k)}$.

Of course I will start from the density function of $\displaystyle Y_{(k)}$. Then I will set up the integral, from $\displaystyle a$ to $\displaystyle b$, of $\displaystyle y$ times this density function. But how can I solve it?

2. ## Re: Uniform distribution between (a,b): expected value of kth order statistic

Originally Posted by Riccardo
Let $\displaystyle Y_1 ,...., Y_n$ be a random sample from an uniform distribution $\displaystyle U[a,b]$, both positives. Let $\displaystyle Y_{(k)}$ be the kth order statistic (that is the k-th smallest value extracted from that sample) from a sample of $\displaystyle n$ observations.
Find the expected value of $\displaystyle Y_{(k)}$.
Of course I will start from the density function of $\displaystyle Y_{(k)}$. Then I will set up the integral, from $\displaystyle a$ to $\displaystyle b$, of $\displaystyle y$ times this density function. But how can I solve it?