Distribution of Discrete Records
Distribution of Discrete Records
Consider a sequence of independent, identically distributed data values , , …. Assume that the data values are from among the non-negative integers 0, 1, 2, … with no upper bound. A value is a record value (or a high-water mark) if it is the largest value among all the values that have been recorded up to time . Let be the record value, ; define =, that is, the first data value is the record value (or the trivial record). This Demonstration shows the distribution of the record values , , when the data has a Poisson, a negative binomial, or a geometric distribution (the special case of the negative binomial distribution with ).
X
1
X
2
X
i
i
R
i
th
i
i=0,1,…
R
0
X
1
th
0
R
i
i=1,…,5
m=1
The blue curve is the probability density function of the data (for a discrete distribution, the probability density function is often also called the probability mass function). The red curve is the density of the record value currently chosen. The green vertical line represents the approximate median of the distribution of the record value: the probability that the record value is at most this value is at least 0.5 (and the probability that the record value is at most this value minus one is less than 0.5).