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}

Read more about this topic:  Order Statistic

Famous quotes containing the words dealing with, dealing, discrete and/or variables:

    “Mediocre people have an answer for everything and are astonished at nothing. They always want to have the air of knowing better than you what you are going to tell them; when, in their turn, they begin to speak, they repeat to you with the greatest confidence, as if dealing with their own property, the things that they have heard you say yourself at some other place.... A capable and superior look is the natural accompaniment of this type of character.”
    —Eugène Delacroix (1798–1863)

    “You’ve been trying to keep an honest accounting of city monies. You’ve been dealing with politicians. You’ve been standing up for your own rights, haven’t you? Naturally you landed in jail.”
    —Dalton Trumbo (1905–1976)

    “We have good reason to believe that memories of early childhood do not persist in consciousness because of the absence or fragmentary character of language covering this period. Words serve as fixatives for mental images. . . . Even at the end of the second year of life when word tags exist for a number of objects in the child’s life, these words are discrete and do not yet bind together the parts of an experience or organize them in a way that can produce a coherent memory.”
    —Selma H. Fraiberg (20th century)

    “Science is feasible when the variables are few and can be enumerated; when their combinations are distinct and clear. We are tending toward the condition of science and aspiring to do it. The artist works out his own formulas; the interest of science lies in the art of making science.”
    —Paul Valéry (1871–1945)