Fall 2011 Homework 9: Question 5
Consider the following:
Let X ~ Bin(n, p) and B ~ Beta(j, n-j+1), where n is a positive integer and j is a positive integer with $j\leq n$. Show using a story about order statistics (without using calculus) that $P(X\geq j)=P(B\leq p)$. This shows that the CDF of the continuous r.v. B is closely related to the CDF of the discrete r.v. X, and is another connection between the Beta and Binomial.
Solution: Let U1,..., Un be i.i.d. Unif(0, 1). Think of these as Bernoulli trials, where Uj is defined to be "successful" if Uj =< p (so the probability of success is p for each trial). Let X be the number of successes. Then X >= j is the same event as U(j) =< p, so P(X >= j) = P(U(j) =< p).
"Mathematics is the logic of certainty, but statistics is the logic of uncertainty."