Editing Order statistic (section)

== Dealing with discrete variables ==
Suppose <math>X_1,X_2,\ldots,X_n</math> are i.i.d. random variables from a discrete distribution with cumulative distribution function <math>F(x)</math> and [[probability mass function]] <math>f(x)</math>. To find the probabilities of the <math>k^\text{th}</math> order statistics, three values are first needed, namely
:<math>p_1=P(X<x)=F(x)-f(x), \ p_2=P(X=x)=f(x),\text{ and }p_3=P(X>x)=1-F(x).</math>

The cumulative distribution function of the <math>k^\text{th}</math> order statistic can be computed by noting that

:<math>
\begin{align}
P(X_{(k)}\leq x)& =P(\text{there are at least }k\text{ observations less than or equal to }x) ,\\
& =P(\text{there are at most }n-k\text{ observations greater than }x) ,\\
& =\sum_{j=0}^{n-k}{n\choose j}p_3^j(p_1+p_2)^{n-j} .
\end{align}
</math>

Similarly, <math>P(X_{(k)}<x)</math> is given by

:<math>
\begin{align}
P(X_{(k)}< x)& =P(\text{there are at least }k\text{ observations less than }x) ,\\
& =P(\text{there are at most }n-k\text{ observations greater than or equal to }x) ,\\
& =\sum_{j=0}^{n-k}{n\choose j}(p_2+p_3)^j(p_1)^{n-j} .
\end{align}
</math>

Note that the probability mass function of <math>X_{(k)}</math> is just the difference of these values, that is to say

:<math>
\begin{align}
P(X_{(k)}=x)&=P(X_{(k)}\leq x)-P(X_{(k)}< x) ,\\
&=\sum_{j=0}^{n-k}{n\choose j}\left(p_3^j(p_1+p_2)^{n-j}-(p_2+p_3)^j(p_1)^{n-j}\right) ,\\
&=\sum_{j=0}^{n-k}{n\choose j}\left((1-F(x))^j(F(x))^{n-j}-(1-F(x)+f(x))^j(F(x)-f(x))^{n-j}\right).
\end{align}
</math>