order statistics corresponding to a sample of size $N$. The PDF $f_{(k,N)}(y)$ is then calculated by differentiating $F_{(k,N)}(y)$ with respect to $y$.

Marginal distribution of minimum order statistic {#subsec:marginal_distribution}
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To further illustrate the problem, consider fist an example in which we have drawn $N=5$ independent random variables $\{y_i\}_{i=1}^5$ each from a common distribution with PDF $f(y)$. Assume that we are interested in the PDF of the first order statistic, $f_{(1,5)}(y)$. The CDF $F_{(1,5)}(y)$ is defined as $P(y^{(1)} < y)$. We note that the minimum order statistic $Y^{(1)}$ would be less than $y$ if at least $1$ of the random variables $y_1, y_2, y_3, y_4, y_5$ are less than $y$. In other words, we need to count the number of ways that can happen such that at least one random variable is less than $y$. This leads to a binomial probability calculation. The ’success’ is considered to be the event $\{y_i < y\}$, $i = 1$ and we let $\zeta$ denote the number of successes in five trials, then $$\begin{aligned}
F_{(1,5)}(y) &= P(y_{(1)}<y) = P(\zeta=1)+\ldots + P(\zeta=5),\\
f_{(1,5)}(y) &= \frac{\,d}{\,dy}F_{(1,5)}(y).\end{aligned}$$ To generalize the example, let $y_{(1)} < y_{(2)} < \ldots < y_{(N)}$ be the order statistics of $N$ independent observations 