Discretizing the probability distribution

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alpedhuez
alpedhuez 2020-7-26
回答： Bruno Luong 2020-7-26
I have a probability distribution f on [0,1]. I want to discretize f on (0,0.1,0.2,...1). I see
"What you could do instead, in order to approximate it to a discrete probability space, is to define some points (x_1, x_2, ..., x_n), and let their discrete probabilities be the integral of some range of the PDF (from your continuous probability distribution), that is P(x_1) = P(X \in (-infty, x_1_end)), P(x_2) = P(X \in (x_1_end, x_2_end)), ..., P(x_n) = P(X \in (x_(n-1)_end, +infty))"
(https://stackoverflow.com/questions/57442670/approximate-continuous-probability-distribution-in-matlab)
Is there any implementation of this idea?
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John D'Errico
John D'Errico 2020-7-26
编辑：John D'Errico 2020-7-26
And what do you think your question asked? That is, it seems to be to find the points x1,x2,...
You don't recognize what was said there as locating the corresponding quantiles of that distribution?
If all you want to do is to find the probbilities for a fixed set of x_i, then all you need to do is compute the integral under the PDF, and this is given directly by the CDF. In any event, this is fairly easy to do.
alpedhuez
alpedhuez 2020-7-26
(x_1, x_2, ..., x_n), are the points in the domain of PDF.

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Bruno Luong
Bruno Luong 2020-7-26
To generate 1D random variable with a prescribed continuous PDF, one need 2 things:
  1. able to integrate pdf (called cdf)
  2. able to invert it
If you know how to do 1 and 2, then problem solves.
If you know to do 1 and not 2, then you can use INTERPOLATION in place invertion, after selecting a set of intervals that cover the domain. I think using 'pchip' is the best since it preserves monotonic and also more accurate than the default 'LINEAR'.
If you don't know to do 1 and 2, but knowing evaluate PDF, then using cumulative sum, trapz rule, simpson rule to replace the continuous CDF by discrete sum, then do the interpolation as above.
In this thread I post some code for the later case. Feel free to modify the code, for example experiment with other interpolation method ('pchip') and integration method (replace cumsum with appropriate higher integration rule).

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