Order statistics of the uniform distribution

Suppose we obtain n measurements of a random variable X following a certain distribution and order them from smallest to largest. We could repeat many times this experiment of measuring n times and sorting. We would get each time a random value for the smallest measurement in the series, a random value for the second smallest measurement in the series, a random value for the third smallest measurement in the series, and so on. Each of these ordered values is a random variable X_(1) , X_(2) , X_(3) , ... , X_(k) , ... , X_(n). We ask ourselves what distribution each random variable follows (in general, the distribution of the k-th order statistic). In this video, we solve this for measurements sampled from a uniform distribution U(a,b) and study the connection to the beta distribution in the case of U(0,1).