Distribution of Records

distribution of the data

UniformDistribution[{0,1}]

1

1

show conditional density

current record

0.5

Consider a sequence of independent, identically distributed, continuous data values

X

1

,

X

2

, …. A value

X

i

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

i

. Let

R

i

be the

th

i

record value,

i=0,1,…

; define

R

0

=

X

1

, that is, the first data value is the zeroth record value (or the trivial record). This Demonstration shows the distribution of the record values

R

i

,

i=1,…,10

, for some distributions of the data.

The blue curve is the probability density function of the data. The light red curves are the probability density functions of the record values, the dark red curve is the density of the record value currently chosen, and the green vertical line is the mean of this distribution. We can also study the conditional distribution of the next record given the current record.