Generate pdf distribution with bias

I'm curious if anyone knows how to bias a probability density function. For example, I want to create a pdf from vector A, but I want to bias it according to vector B (without changing the values in A). So if A = [1 2 3 4 5] and B = [1 1 1 1 0.5], the 5th element of A would count half as much as the rest when computing my pdf.

In that case, a simple work-around is to re-create the other elements in A to weight them twice as much, so A' = [1 2 3 4 5 1 2 3 4]. However, this gets more difficult if the biases are not easy integers. If my weighted vector B = [.4265 1 .9361 1 .1532], it would not be easy to simply add more elements. The only solution I can think of is to find the closest common denominator for the biases and add elements accordingly. Seeing as I will be repeating this with much larger vectors and changing biases, a better way would be nice. I haven't been able to find any toolboxes or function options that offer this. If anyone knows a simpler solution I would be most appreciative. Thanks!

Answer 1

[John D'Errico]

Simpler than you apparently think. Even trivial.

A = 1:5;
B = [1 1 1 1 0.5];
B = cumsum([0,B])/sum(B);

N = 10000;
[~,~,bin] = histcounts(rand(N,1),B);
X = A(bin);
hist(X,100)

I think you should see that the above code will work regardless of the weights, integer, non-integer. So for your other example...

B = [0.4265 1 .9361 1 0.1532]
B = cumsum([0,B])/sum(B);
N = 1000000;
[~,~,bin] = histcounts(rand(N,1),B);
X = A(bin);
hist(X,100)

With a much larger sample, you can see the sample accurately follows the desired distribution.

[Tyler]

Thanks! Does it matter what N is? Just a high enough number to get an accurate spread?

[Tyler]

Thanks John. I notice my results change depending on the value of N. Is there a rule of thumb on that value? Should it be a multiple of the size of my B vector, or just a big enough number to ensure an accurate weighting?