# Distribution of longest streak of consecutive (non-)birthdays in birthday problem

Given the setting of the birthday paradox ($n$ people with birthdays randomly drawn from the uniform distribution of 365 days), let us call the days on which at least one person was born birthdays and all other days non-birthdays. What can be said about the distribution of

a) The length in days of the longest streak of birthdays;

b) The length in days of the longest streak of non-birthdays?

Is there an analytical solution to, for example, the expected value of either of these?

(You can choose whether to break all streaks when the year changes; I think insightful analysis is more interesting than the precise formulation of the problem.)

• Because this question has nothing essential in common with the Birthday Paradox, I have removed that tag. After all, it's not concerned about the chances of collisions in the sample--it's concerned about a completely different kind of event. – whuber Sep 30 '16 at 16:25
• Is the year assumed to be cyclic? In other words, if you see birthdays on days 364,365,1,2 , is that a consecutive streak? – Alex R. Sep 30 '16 at 18:16
• I think my formulation of the question addresses that: "You can choose whether to break all streaks when the year changes; I think insightful analysis is more interesting than the precise formulation of the problem." – Sami Liedes Oct 1 '16 at 19:58
• A continuous version of the problem is: Observe $n$ independent uniform random variables on $(0,1)$, sort them as the order statistics $U_{(1)} < U_{(2)}< \dotsm < U_{(n)}$, compute the $n-1$ spacings $U_{(i)}-U_{(i-1)}$, what is the distribution of (expectation of) the largest spacing? That is answered at: stats.stackexchange.com/questions/162560/… – kjetil b halvorsen Mar 25 '18 at 12:33