Consider a collection of n individuals. . Assume that each person’s birthday is equally likely to be any of the 365 days of the
year and also that the birthdays are independent. For i < j, let Ai,j denote the event that persons i and j have the same birthday, and let X be the number of pairs of people whose birthdays are the same, i.e., the number of Ai,j that occur.(b) Compute E[X], and find the smallest value of n for which E[X] > 1. Describe in words what this means. (A sentence or two is fine.)
1 answer:
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We know:
and
These expressions have an term in common. We can combine them if we can find forms of the expressions in terms of
:
Now, because (by the reflexive property), we can write:
This is our answer, A.