Calculating the new beta parameters given two variables

I have two beta distributed random variables with parameters, mu1, alpha1 and beta1 and mu2, alpha2, beta2 for variables A and B. I would like to generate random variable C witth parameters mu3, alpha 3, and beta 3 to use in another computation. How can I go about tthis?

[Torsten]

If you have the Statistics and Machine Learning Toolbox, you can use

https://de.mathworks.com/help/stats/betarnd.html

[Lewis Waswa]

Thank you @Torsten. isn't that betarnd function generates the random numbers for given parameters A and B of a beta distribution? At least that is the way I understand it. However, I am keen to know, for two random variables K1 and K2, with parameters for K1=(mu1,a1,b1) and K2 (mu2, a2, b2), how do you calculate the parameters of the sum of variables K1 and K2 - or alternatively, how do you get the parameters of their joint distribution? We can call these parameters mu3, a3 and b3.

[the cyclist]

@Lewis Waswa, your question and comment are confusing to me. Here are my points of confusion, that it might help to clarify:

• Your original question doesn't mention anything about the third variable being a sum (or any relationship at all) of the first two variables.
• A beta distribution is defined by two, not three parameters. Are you just saying that you also know the mean, and you are certain it is consistent with a and b?
• When you say you "have" two random variables, do you mean you have samples that have been fit, or do you mean you know the parameters?
• There is a (big) difference between [1] the joint distribution of K1 and K2, and [2] the distribution of the sum of K1 and K2. Which do you actually need?
• Are K1 and K2 independent from each other, or are they correlated?
• Are you trying to calculate the parameters of the third variable, or do you just need to generate random values?

I think you may have not thought carefully about these important details, but at the very least you have not told us these important details.

[Lewis Waswa]

Thank you @the cyclist and @John D'Errico. Please allow me to clarify @the cyclist. I have clarified the first concern using my reply to @Torsten. sorry for not being clear. On tthe second question, thank you for the correction, however I do know that the alpha and beta parameters are used. On the third question, i have the parameters for the two random variables, nott the variables themselves. On the 4th question, I would like to know the parameters of the resulting of the sum of k1 and k2. And k1 and k2 are nott correlated. Lastly, I am trying to calculate the parameters of the third variable.

[John D'Errico]

@the cyclist - some people extend the beta to live on different support, instead of [0,1] or [-1,1]. That can result in either a 3 or 4 parameter beta, depending on the bounds chosen. It is just a shift and/or a scale though, nothing more in that.

@Lewis Waswa -

As I said, there is no named distribution that represents the sum of two beta random variables. Wanting one will not make it appear.

Alternatively, if you like, there IS a distribution, but it requires 4 parameters (not to mention the support) to represent that distribution. They are the original beta parameters of each of the parents. But that is essentially what I suggested in my answer. If you want to find the sum of two betas, then just start by sampling the two betas, then add them together. Anything else you would do would only be an approximation, and that approximation would often be a terribly poor one. Whereas, my suggestion is in fact the exactly correct distribution, because there are no approximations needed.

Answer 1

[John D'Errico]

I saw this question when you first posted, but it was not at all clear what was your question. Now I see you want to compute the distribution of the sum of two beta random variables, as a function of their parameters. This is of course a trivial thing to do for Normally distributed random variates, but far less so when the variables are not one of the easy to deal with distributions. And a beta distribution can be pretty nasty, since some sets of parameters will yield a bimodal distribution. You can do some reading online, but I think you will not find any simple analytical result.

https://math.stackexchange.com/questions/965226/sum-of-two-beta-distributed-random-variables

http://www.mathstatica.com/Sumof2Betas/

In fact, you MIGHT try to claim that the sum should be another beta distriibution, but that is clearly false. (It MIGHT not be a terribly bad approximation for some sets of parameters, when both of the parents are simple unimodal forms. Remember that the support of the distribution will change, but that is just a scaling thing.) But consider the case where each beta distribution has alpha=beta=1. (This is the case where a beta RV reduces to a uniform distribution.) But the sum of two uniform random variables has a triangular distribution. And there is no set of beta parameters that represents a triangular distribution. So it MUST fail, if you think the sum might also be another beta random variable.

And you might decide to try using various statistical tolerancing techniques to compute the distribution of the sum. You could use the Pearson or Johnson family, then to generate the result. But those distributions pretty much all have infinite support as I recall. ANd while that would not be terrible for some sets of parameters, again, there are a wide variety of shapes you can start with for each of the beta parents, and the sum will not fit into any simple distribution in general.

If your goal is simply to generate a random variable that is the sum of two beta random variables however, then surely this is trivial. There is no need to compute new beta parameters. Just use the parent beta parameters, and make two calls to betarnd. What could be easier? The nice thing is, this will have EXACTLY the correct distribution, even though you need not have even a clue as to the true distribution. In the end, I think you are looking for a complicated solution to a trivially easy problem.

[Lewis Waswa]

@John D'Errico. This is insightful. Indeed with a1 and b2, I can generate k1 and the same applies to k2 where I have a2 and b2. But tthen I would be stuck with two variables.

% sample code
k1=betarnd(a1,b1,[400,50])
k2 =betarnd(a2,b2,[400,50])
k3

How do I then obtain the k3, is it a simple sum of the same? If that be the case then I could simply obtain the beta parameters of the resulting sum of k1 and k2.

[John D'Errico]

a1 = 2;b1 = 3;
a2 = 1.5;b2 = 4;

k1 = betarnd(a1,b1,[1,100000]);
k2 = betarnd(a2,b2,[1,100000]);

k3 = k1 + k2;

histogram(k3)

That is all you need to do. As long as you are not talking about a mixture distribution, a subtly different thing. But you have never said anything about mixture distrbutions, just the sum of random variables. (Actually, that is also not difficult to do.) If you just want it in one call, then just do this:

betasumrnd = @(a1,b1,a2,b2,sz) = betarnd(a1,b1,sz) + betarnd(a2,b2,sz);

As I said, just a 4 parameter distribution.