How can I combine probability distribution objects?

Are there operators probability distribution objects? For example,

>>load hospital
>>xA = hospital.Weight;
>>pdA = fitdist(xA,'Normal')

Suppose I have a similar second record

    >>hospitalB

where there are k times as many patients. Then I could write

>>pdboth = fitdist([xA;xB],'Normal');

but if I've only got the pd and weighting - or want to do some other weighting - I'd like to do something like

>>pdboth = pdA + k*pdB;

Does this sort of operator exist, or do I need to convert to a pdf and then fit a new distribution?

[Tom Lane]

What are you expecting from the addition? Are you looking for a mixture of Gaussians (perhaps a bimodal distribution with one peak from each distribution)?

Answer 1

[Arnab Sen]

I understand that you would like to achieve pooled weighted distribution of two distributions. There is no in-built functions in MATLAB to achieve this goal. However, it is possible to calculate it manually.

Consider following MATLAB code snippet for illustration:

>>muA=pdA.mu;
>>sigA=pdA.sigma;
>>lA=length(xA);
>>muB=pdB.mu;
>>sigB=pdB.sigma;
>>lB=length(xB);
>>pdboth=fitdist([xA;xB],'Normal');
%Calculate mean and standard deviation programmatically
>>muC=(muA*lA+k*muB*lB)/(lA+k*lB);
>>varC=((lA-1)*sigA^2+k*(lB-1)*sigB^2)(lA+k*lB-(k+1));
>>sigC=sqrt(varC);
>>pdboth.mu=muC;
>>pdboth.sigma=sigC;

In simple words we are calculating 'pdboth' as the distribution of xC=[xA;xB;xB;...ktimes]. You can also simply create a vector 'xC' as above and compute 'pdboth'.

For more details of the calculation as shown in the code, refer to the following link:

https://en.wikipedia.org/wiki/Pooled_variance