Order Statistic - Dealing With Discrete Variables

Dealing With Discrete Variables

Suppose are i.i.d. random variables from a discrete distribution with cumulative distribution function and probability mass function . To find the probabilities of the order statistics, three values are first needed, namely

The cumulative distribution function of the order statistic can be computed by noting that

\begin{align} P(X_{(k)}\leq 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}

Similarly, is given by

\begin{align} P(X_{(k)}< 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}

Note that the probability mass function of is just the difference of these values, that is to say

\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}

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