I’ve been studying probability on my own for a little while now, and all the distributions I’ve come across so far have nice, intuitive explanations for what they are. E.g., an exponential variable is the time-to-arrival for a Poisson event, a gamma variable is time-to-arrival for n Poisson events, geometric r.v. the probability of n Bernoulli trials before a success… but now I’ve got the Beta distribution with no intuitive explanation for what it means or how it works.
Rather than writing out the formulas, I’ll link to the relevant Wikipedia page.
So you math guys: can you explain how this distribution might apply to the real world? How is it used in math, science or finance?
A beta distribution, where the expected value is a/(a + b) can be understood readily for positive integer values, I think. Let’s say that there are two guys, who each have a random number generator that generates values from in the same range with the same odds for any given subset of that range. It really doesn’t matter what the exact distribution is, as long as they’re identical.
The first guy generates a random values, and the second guy generates b random values. Each guy takes the highest value that they have and then compares it to the other guy’s. The odds that the first guy’s value is higher than the second guy’s value is a/(a + b). It just so happens that this equation works for any numbers, not just positive integer values.
Oldguy : Are you sure? I thought that the binomial distribution approached the normal distribution–or at least, the normal is a good approximation of the binomial when np and n(1-p) are greater than 10. Or do you mean something else by the ‘limit of the distribution’?
Punoqllads: that’s an interesting application, only I’m not sure how to extend that concept in my mind from discrete to continuous (Bob generates 4.6 numbers? :smack: ) except, perhaps, through interpolation.
Dang, one more point. To understand a = 4.6, think of it as a = 4 for 40% of the time, and a = 5 for 60% of the time. So a = 4.6 and b = 2.1 means that:
36% of the time, the first guy generates 4 numbers and the second guy generates 2 numbers.
4% of the time, the first guy generates 4 numbers and the second guy generates 3 numbers.
54% of the time, the first guy generates 5 numbers and the second guy generates 2 numbers
6% of the time, the first guy generates 5 numbers and the second guy generates 3 numbers.
Can’t be. As n [symbol]® ¥[/symbol], the pdf of the binomial distribution becomes a function of one variable, and the pdf of the beta distribution is a function of two variables.
It may be that you get a special case of the beta distribution, but not the general case.
Sorry that was very badly explained. Yes in the limit a binomial distribution is normal. So let’s try again
Let B(n; N, p) be the cumulative binomail distribution; that is B(n; N, p) is the probability of n or fewer successes out of N trials when the probability of each success is p.
Let J(x; a, b) denote the cumulative beta distirbution.; that is J(x; a, b) is the probablity that a beta variable with parameters a and b is less than x.
Then B(n; N, p) = J(1-p; N-n+1, n)
So if we think of the number of success, n, as random, we want the binomial distribution. If we want to think of the number of successes as observed and the probability of each success as unknown, then we want the beta distribution.
Interesting. But–how does this relate to the negative binomial distribution? IIRC negative binomial is used when the number of successes is fixed, and you’re finding the probability of the variable {number of trials}–isn’t that the same as what you just said?