Simulate samples from a joint cumulative distribution function?

Question

I have a joint density function for two independent variables X and Y. And I now want to sample new x,y from this distribution.

What I believe I have to do is to find the joint cumulative distribution and then somehow sample from it. I kinda know how to do this in 1D, but I find it really hard to understand how to do it in 2D.

I also used the matlab function cumtrapz to find the cumulative distribution function for the above pdf.

Just to be clear, what i want to do is to sample random values x,y from this empirical distribution.

Can someone please point me in the right direction here?!

EDIT: I have data values and I use [pdf bins] = hist3([N Y])

I then normalize the pdf and do

cumulativeDistribution = cumtrapz(pdfNormalize)

And yes (to the comment below) X,Y are suppose to be independent.

If your variables are independent, your joint distribution is simply the product of marginals. Are you sure that's what you meant? — Memming, Apr 18 '12 at 15:00
Do you have analytic formulae for your joint density? Or are you estimating from samples? What are you using to estimate the joint? — Memming, Apr 18 '12 at 15:05

score 6 · Accepted Answer · edited Jun 05 '17 at 15:35

6

If you know how to sample a distribution in 1D then you can extend it to 2D. Create the marginal distribution for X. Take a sample from that, say X₁. Then in your 2D distribution fix one variate X=X₁ and sample for Y, i.e., sample Y from 1D distribution f_XY(X₁, Y).

edited Jun 05 '17 at 15:35

Peter O.

32,158
14
82
96

answered Apr 18 '12 at 12:20

Mohsen Nosratinia

9,844
1
27
52

1

This works! :) The same is almost done in this very nice .m-file from the MATLAB file exchange: http://www.mathworks.com/matlabcentral/fileexchange/35797-generate-random-numbers-from-a-2d-discrete-distribution – Groot Apr 19 '12 at 08:05
This is correct but would you have a solution when only relying on the cdf? – Xi'an ні війні Jun 17 '22 at 14:24

Kerry M. Soileau · Answer 2 · 2015-01-31T16:47:18.227

3

Given a joint distribution in say, two random variable X and Y, you can compute a CDF for X alone by summing over all possible values of Y, i.e. P(X<=x)=Sum[P[X=x_i and Y=y_j],{x_i<=x and all values of y_j}]. Once you have P(X<=x) in hand, there are well-known methods for sampling a value of X, let's call it a. Now that you have a, compute P(Y<=y given X=a)=Sum[P[X=a and Y=y_j],{y_j<=y}]/Sum[P[X=a and Y=y_j],{all values of y_j}]. Sample Y using the same approach that gave you X, yielding Y=b. Done.

The same approach generalizes to more that two random jointly distributed random variable.

edited Jan 31 '15 at 16:47

answered Jan 16 '15 at 15:07

Kerry M. Soileau

51
5

This answer was very helpful to me! This put me on the right path, and I was able to implement this into a useful sampling method for the Kent distribution. – amicitas Mar 12 '22 at 03:48

Simulate samples from a joint cumulative distribution function?

2 Answers2

Linked