Approximations to the birthday problem with unequal occurrence probabilities and their application to the surname problem in Japan

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 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.This research is supported in part by Grant-in-Aid for Scientific Research of the Ministry of Education, Science and Culture under the contact number 01540141 and 02640057.