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APPROXIMATIONS TO THE BIRTHDAY PROBLEM WITH

UNEQUAL OCCURRENCE PROBABILITIES AND THEIR

APPLICATION TO THE SURNAME PROBLEM IN JAPAN

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SHIGERU MASE

*Faculty of Integrated Arts and Sciences, Hiroshima University,*

Naka-ku, Hiroshima 730, Japan
(Received July 4, 1990; revised September 2, 1991)

**Abstract.**
Let *X*_{1},*X*_{2},....,*X*_{n} be iid random variables with
a discrete distribution {*p*_{i}}_{i=1}^{m}.
We will discuss the coincidence probability *R*_{n}, i.e., the
probability that there are members of {*X*_{i}} having the same value.
If *m* = 365 and *p*_{i} \equiv 1/365, this is the famous birthday problem.
Also we will give two kinds of approximation to this probability.
Finally we will give two applications.
The first is the estimation of the coincidence probability of
surnames in Japan.
For this purpose, we will fit a generalized zeta distribution
to a frequency data of surnames in Japan.
The second is the *true* birthday problem, that is, we will
evaluate the birthday probability in Japan using the actual (non-uniform)
distribution of birthdays in Japan.

*Key words and phrases*:
Birthday problem, coincidence probability,
non-uniformness, Bell polynomial, approximation, surname.

**Source**
( TeX ,
DVI ,
PS )