how can I effectively compute expected value by histogram approximation of probability desnsity function

What is the proper way to compute effectively (fast) the expected value E(x) in a case when I have approximation of probability desity function f(x) by probability normalized histogram?

Is there (FEX) any code available?

[the cyclist]

In what form do you have the approximate pdf? Is it a MATLAB function? Or is it a series of discrete values at specific x locations? Or something else? Can you upload your input data?

[Michal]

It is structure of discrete values similar to matlab histogram object.

Answer 1

[the cyclist]

There might be some nuances in the numerical integration, but here is the basic idea. You need to approximate the integral of x over the pdf.

% For reproducibility
rng default

% Simulated data -- normal centered on x=5.
N = 1000000;
x = 5 + randn(N,1);

% Get the probability density function. (You have these values already?)
[pdf_x,xi] = ksdensity(x);

% The bin width. (In this case, they are all equal, so I just take the first one.)
dx = xi(2) - xi(1);

% Calculate the total probability. (It should be 1.)
total_probability = sum(pdf_x*dx)

% Calculate the mean, which is the expected value of x.
mean_x = sum(xi.*pdf_x*dx)

[Michal]

Yes, but your example is too simple. My histogram probability density approximation data are defined more general (BinEdges, NumBins, etc) with non-equidistant BinWidths.

Not clear to me how to work with x values outside of BinLimits, for example.

I would like to avoid of any smooth density estimation, because my histograms are very rough. In this case is not clear how to choose proper Bandwidth parameter of ksdensity function.

[the cyclist]

The ksdensity was only for me to simulate something like what you probably had -- a discrete vector of locations and densitiies. You shouldn't need to do any smoothing, etc.

If you have irregular bin widths, all you need to do is the numerical integration of x * pdf * width, as I did.

Can you post your data?

[Michal]

OK ... thanks for basic info. I think the "nuances" of integration will be my main problem.

I will try to use matlab "discretize" function to find pdf approximation at my histogram points and then integration over x * pdf * width.

[the cyclist]

You might be able to use the trapz function.