from a continuous distribution with cumulative distribution function $F(y)$ and probability density function $f(y)=F'(y)$. The marginal PDF $f_{(1,N)}(y)$ of the minimum order statistic can be obtained by considering the event $\{Y_i \leq y\}, i = 1$ as a “success,” and letting $\zeta$ = the number of such successes in $N$ mutually independent trials. $\zeta$ is a binomial random variable with $N$ trials and probability of success $P(y_i\leq y)$. Hence, the CDF of the minimum order statistic is given by,

$$\begin{aligned}
    F_{(1,N)}(y)=\sum_{n=1}^{N}P(\zeta=n).
    \label{eq:cdf_order_minimum}
    \end{aligned}$$

Noting that the probability mass function of this binomial distribution is given by, $$\begin{aligned}
    P(\zeta=n) = \begin{pmatrix}N\\n\end{pmatrix}[F(y)]^n[1-F(y)]^{N-n}.
    \label{eq:pmf_order_minimum}
    \end{aligned}$$ Substituting  into  and taking the last term out of the sum, we get $$\begin{aligned}
    F_{(1,N)}(y)=\sum_{n=1}^{N-1}\begin{pmatrix}N\\n\end{pmatrix}[F(y)]^n[1-F(y)]^{N-n}+[F(y)]^N.
    \label{eq:cdf_order_minimum_2}
    \end{aligned}$$

Differentiating  with respect to $y$ gives a telescoping sum of the form, $$\begin{aligned}
f_{(1,N)}(y)&=\sum_{n=1}^{N-1}\frac{N!}{(n-1)!(N-n)!}[F(y)]^{n-1}f(y)[1-F(y)]^{N-n}\nonumber\\ &+ \sum_{n=1}^{N-1}\frac{N!}{n!(N-n-1)!}[F(y)]^n[1-F(y)]^{N-n-1}(-f(y)) \nonumber\\&+ N[F(y)]^{N-1}f(y),\end{aligned}$$ in which, except the first term, all other terms cancel each other out. Hence, the marginal probability density function of the minimum order statistic of a set of $N$ independent and identically random variables with common CDF $F(y)$ and PDF $f(y)$ is given by, $$\begin{aligned}
f_{(1,N)}(y) = Nf(y)\left(1-F(y)\right)^{N-1}.
\label{eq:density_order_minimum}\end{aligned}$$

Marginal distribution of general order statistic {#subsec:marginal_distribution_of_general_order_statistic}
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The marginal PDF $f_{(k,N)}(y)$ of the 