The binomial distribution statistics
For large and we can use Stirling's approximation. To find , set this expression to 0 and solve for ,. We can now find the terms in the expansion. Since each term is of order smaller than the previous, we can ignore terms higher than , so. The binomial distribution is therefore approximated by a normal distribution for any fixed even if is small as is taken to infinity. If and in such a way that , then the binomial distribution converges to the Poisson distribution with mean.
Let and be independent binomial random variables characterized by parameters and. The conditional probability of given that is. Probability, Random Variables, and Stochastic Processes, 2nd ed. The Art of Scientific Computing, 2nd ed.
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You can read more about it at Combinations and Permutations. The probabilities for "two chickens" all work out to be 0. But we need to include that there are three such ways it can happen: That was a lot of work for something we knew already, but now we have a formula we can use for harder questions. Moral of the story: Have a play with the Quincunx then read Quincunx Explained to see the Binomial Distribution in action.
It is not symmetrical! What is the expected Mean and Variance of the 4 next inspections? There are relatively simple formulas for them.
They are a little hard to prove, but they do work! So we can expect 3.