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:

    “The bulk of mankind believe in two gods. They are under one dominion here in the house, as friend and parent, in social circles, in letters, in art, in love, in religion; but in mechanics, in dealing with steam and climate, in trade, in politics, they think they come under another.”
    —Ralph Waldo Emerson (1803–1882)

    “The public history of modern art is the story of conventional people not knowing what they are dealing with.”
    —Robert Motherwell (1915–1991)

    “The mastery of one’s phonemes may be compared to the violinist’s mastery of fingering. The violin string lends itself to a continuous gradation of tones, but the musician learns the discrete intervals at which to stop the string in order to play the conventional notes. We sound our phonemes like poor violinists, approximating each time to a fancied norm, and we receive our neighbor’s renderings indulgently, mentally rectifying the more glaring inaccuracies.”
    —W.V. Quine (b. 1908)

    “The variables of quantification, ‘something,’ ‘nothing,’ ‘everything,’ range over our whole ontology, whatever it may be; and we are convicted of a particular ontological presupposition if, and only if, the alleged presuppositum has to be reckoned among the entities over which our variables range in order to render one of our affirmations true.”
    —Willard Van Orman Quine (b. 1908)