# Poisson Paradigm the distribution of the number X

Poisson Paradigm the distribution of the number X of birthday matches is approximately Pois(Lambda) where Lamda = (m choose 2) multiplied by 1/365.where m is people. If m=23 then lambda = 253/365 and 1-e to the negative lamda shows approximately 50% chance that at least one of them has the same birthday.a) Using the same logic above. SHOW how many people are need to have a 50% chance that there are two people who not only were born on the same day but also were born at the same hour for instance same hour means between 2pm and 3pm would be considered the same hour.b) Considering that only 1/24 of pairs of people born on the same day were born at the same hour why isnt the answer to (a) approximately (24 multiplied by 23)? Explain this intuitively and give a simple approximation for the factor by which the number of people needed to obtain probability p of a birthday match needs to be scaled up to obtain probability p of a birthday-birthhour match.c) With 100 people there is a 64% chance that there are 3 with the same birthday (according to R using pbirthday(100 classes=365 coincident=3) to compute it). Provide two di?erent Poisson approximations for this value one based on creating an indicator r.v. for each triplet of people and the other based on creating an indicator r.v. for each day of the year. Which is more accurate?

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