Re: and sampling a distribution

*To*: mathgroup at smc.vnet.net*Subject*: [mg110566] Re: and sampling a distribution*From*: Bill Rowe <readnews at sbcglobal.net>*Date*: Sat, 26 Jun 2010 03:09:18 -0400 (EDT)

On 6/25/10 at 7:27 AM, stone at geology.washington.edu (John Stone) wrote: >I am trying to use RandomReal[ ] to sample from bins of different >widths that span the interval 0 - 1. The bin widths represent the >weights I'm assigning to a family of trial solutions in an >optimization problem. The aim is to sample the solutions in >proportion to their weights using a uniform distribution of random >numbers generated by RandomReal[ ]. >For a simple example, however, suppose there are 10 equally weighted >solutions. My selection process would use some code that looks >like: >weights = Table[0.1, {10}]; >bins = Accumulate[weights]; >Select[bins, (# >= RandomReal[] &)][[1]] Rather than RandomReal you should be using RandomChoice. Specifically, RandomChoice[weights->bins,10] will return a list of 10 values with the desired distribution. This can be seen by doing: Histogram[RandomChoice[weights -> bins, 1000]] and note with equal weights and equally spaced bins of size 0.1, the following is equivalent RandomInteger[{1,10}]/10//N >Assuming the result of RandomReal[ ] is uniformly distributed, I >expected this to return 0.1 as frequently as it returns 0.5 or 1, No, this isn't correct. The value 0.1 will be returned whenever RandomReal returns a value greater than or equal to 0.1 but less than 0.2 which should happen 10% of the time. But the value 1 will be returned only if RandomReal returns the value 1 which will happens with probability near 0. Now consider what happens when RandomReal returns a value greater than 0.1 but less than 0.3. This will occur ~20% of the time. And your selection criteria will return 0.2 as the first value in the list of selected values. That is 0.1 occurs with probability 10%, 0.2 occurs with probability 20% an 1 occurs with very low probability (near 0). So, it is clear the distribution with this selection criteria cannot be flat as you were expecting. I haven't worked out the probability for the other values in the list. I think the above is sufficient to show the selection criteria you have used will not return uniform deviates.