Marginal pdf of Order Statistics

Theorem

Let

• $X_1,\dots ,X_n$ : Random sample with pdf $f(x),a<x<b$
• $Y_1,\dots ,Y_n$ : Order statistics of $X_1,\dots ,X_n$

Then marginal pdf of $Y_1,\dots ,Y_n$ is given by:

• $Y_1$

$$g_1\left(y_1\right)=\{\begin{array}{ll}n{\left[1-F\left(y_1\right)\right]}^{n-1}f\left(y_1\right),& a<y_1<b\\ 0,& \text{elsewhere}\end{array}$$

• $Y_k$

$$g_k\left(y_k\right)=\{\begin{array}{ll}\frac{n!}{(n-k)!(k-1)!}{\left[1-F\left(y_k\right)\right]}^{n-k}{\left[F\left(y_k\right)\right]}^{k-1}f\left(y_k\right)& a<y_k<b\\ 0& \text{elsewhere}\end{array}$$

• $Y_n$

$$g_n\left(y_n\right)=\{\begin{array}{ll}n{\left[F\left(y_n\right)\right]}^{n-1}f\left(y_n\right),& a<y_n<b\\ 0,& \text{elsewhere}\end{array}$$

The joint marginal pdf of $Y_i$ and $Y_j$ is given by:

$$g_{ij}\left(y_i,y_j\right)=\{\begin{array}{ll}\frac{n!}{(i-1)!(j-i-1)!(n-j)!}{\left[F\left(y_i\right)\right]}^{i-1}{\left[F\left(y_j\right)-F\left(y_i\right)\right]}^{j-i-1}{\left[1-F\left(y_j\right)\right]}^{n-j}f\left(y_i\right)f\left(y_j\right)& a<y_i<y_j<b\\ 0& \text{ elsewhere }\end{array}$$