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

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Michal Kvasnicka on 22 Oct 2019
Commented: the cyclist on 22 Oct 2019
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 on 22 Oct 2019
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 Kvasnicka on 22 Oct 2019
It is structure of discrete values similar to matlab histogram object.

the cyclist on 22 Oct 2019
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)

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the cyclist on 22 Oct 2019
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.
Michal Kvasnicka on 22 Oct 2019
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 on 22 Oct 2019
You might be able to use the trapz function.