Discretizing the probability distribution

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?

[John D'Errico]

Not sure where the problem lies. One (short) line of code, based on the inverse CDF? Something like:

X = norminv(0.1:.1:.9);

If you lack the stats toolbox, then tools like erfinv are a direct substitute for norminv. You may need to do some reading in Abramowitz and Stegun for many of the common distributions, since a few are based on a transformation of the the gamma, and matlab provides an inverse gamma and inverse beta.

If you don't have the inverse CDF, then you could use a root finder, applied to the CDF.

If you have only the PDF, and not the CDF, then you use a rootfinder applied to a integral of the PDF.

[alpedhuez]

I want to discretize the distribution. If I understand, norminv will give the quantile of the distribution.

[John D'Errico]

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]

(x_1, x_2, ..., x_n), are the points in the domain of PDF.

Answer 1

[Bruno Luong]

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).