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Here's another formulation of a birthday problem: given $n$ people, and $m$ days, how to calculate the expected number of people having a birthday on any single day? I.e., given a random variable $X$ denoting a number of people with a birthday on a given date, how to find $\mathbb{E}[X]=\sum_{k=0}^{n}k\mathbb{P}[X=k]$?

Henry
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kck
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1 Answers1

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Let $X_i = 1$ if person $i$ has a birthday on fixed day $b$, and $0$ otherwise. We want to know the value of $X = X_1 + X_2 + X_3 + ... + X_n$ on average.

Then by linearity of expectation:

$$E[X] = E\left[\sum_{i=1}^{n}X_i\right] = \sum_{i=1}^{n} E[X_i] = \sum_{i=1}^{n} \dfrac{1}{m} = \dfrac{n}{m}$$

Marcus Andrews
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  • hi Marcus, thanks, I've realized the answer is indeed trivial. Actually I've made a mistake while formulating... I was looking for an expected birthday collision size, so looking only on the expected number of people on a date, where $X\geq 2$. Do you have any advice on this one, or should I rather post a separate question? – kck May 19 '17 at 16:25
  • @mck Probably better to post a separate question / clarify exactly what it is you're looking to find, if the expected number of people who have birthday $b$ (for fixed $b$) is not what you're looking for. – Marcus Andrews May 19 '17 at 16:26
  • No, what I'm looking for is: given any day with a birthday collision (two or more people have a birthday), what is the expected number of people with a birthday on this day? Here it is: https://math.stackexchange.com/questions/2288036/birthday-problem-expected-birthday-collision-size – kck May 19 '17 at 16:35
  • see the other questions, I hope I've explained better... – kck May 19 '17 at 16:38