From the binomial theorem, we have
Differentiating (1) reveals
Setting in (2) yields
And we are done!
Beginner2022-01-26Added 12 answers
Skilled2022-01-27Added 437 answers
Proof without derivatives:
Alternate proof via probability theory:
Toss a fair coin n times, find the expected no of heads. Let N be the random variable denoting the number of heads. Then because N is the sum of n bernoulli random variables with probability 1/2. But we also know that N has a binomial distribution. Hence
Rearrange to get your answer.
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