Definition 1: For the binomial distribution the number of successes x is the random variable and the number of trials n and the probability of success p on any single trial are parameters (i.e. constants). Instead we would like to view the probability of success on any single trial as the random variable, and the number of trials n and the total number of successes in n trials as constants.
Let α = # of successes in n trials and β = # of failures in n trials (and so α + β = n). The probability density function (pdf) for x = the probability of success on any single trial is given by
This is a special case of the beta function
where Γ is the gamma function.
Excel Functions: Excel provides the following functions:
BETADIST(x, α, β) = the cumulative distribution function F(x) at x for the pdf given above.
BETAINV(p, α, β) = x such that BETADIST(x, α, β) = p. Thus BETAINV is the inverse of BETADIST.
Excel 2010/2013 provide the following two additional functions: BETA.INV which is equivalent to BETAINV and BETA.DIST(x, α, β, cum) where cum takes the value TRUE or FALSE and BETA.DIST(x, α, β, TRUE) = BETADIST(x, α, β) while BETA.DIST(x, α, β, FALSE) is the pdf of the beta distribution at x (as described above).
Example 1: A lottery organization claims that at least one out of every ten people wins. Of the last 500 lottery tickets sold 37 were winners. Based on this sample, what is the probability that the lottery organization’s claim is true: namely players have at least a 10% probability of buying a winning ticket? What is the 95% confidence interval?
To answer the first question we use the cumulative beta distribution function as follows:
BETADIST(.1, 37, 463) = 98.1%
This represents that organization’s claim is false (i.e. less than 10% probability of success). Thus the probability that the organization’s claim is true is 100% – 98.1% = 1.9%
The lower bound of the 95% confidence interval is
BETAINV(.025, 37, 463) = 5.3%
The upper bound of the 95% confidence interval is
BETAINV(.975, 37, 463) = 9.8%
Since 10% is not in the 95% confidence (5.3%, 9.8%), we conclude (with 95% confidence) that the lottery’s claim is not accurate.