Simulating random variables from a discrete distribution II

Question

I have the following discrete probability distribution where $p$, $q$ and $r$ are known constants:

$P(X=0)=q$, $0<q<1$

$P(X=1)=1-p-q-r$, $0<p+q+r<1$

$P(X=2)=p$, $0<p<1$

$P(X=3)=r$, $0<r<1$

How can I sample from this distribution?

score 2 · Answer 1 · answered May 07 '20 at 11:57

To generate $n$ independent values from this distribution, you can use:

sample(0:3, size = n, replace = TRUE, prob = c(p, 1-p-q-r, q, r))

For example, here we generate $n=10^6$ values and show the sample proportions:

#Generate a large number of values from this distribution
set.seed(1)
p <- 0.11
q <- 0.20
r <- 0.35
n <- 10^6
X <- sample(0:3, size = n, replace = TRUE, prob = c(p, 1-p-q-r, q, r))

#Show sample proportions
table(X)/n

X
       0        1        2        3 
0.109570 0.340265 0.200222 0.349943

How do you do this in SPSS? You may be able to create an SPSS syntax file with the sps extension. — Ad van der Ven, May 07 '20 at 14:49

score 0 · Answer 2 · answered May 07 '20 at 10:39

0

There is a quite simple strategy for discrete distributions such as this with small number of elements in the support:

u <- generate a uniform RV in [0,1]
if u < q: x <- 0
elseif u < 1-p-r: x <- 1
elseif u < 1-r: x <- 2
else x <- 3

This is basically an over simplification of Inverse Transform Sampling.

answered May 07 '20 at 10:39

gunes

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How do you do this in SPSS? You may be able to create an SPSS syntax file with the sps extension. – Ad van der Ven May 07 '20 at 14:49
I don't know SPSS. But, it's so simple that you should be able to implement it in almost every language you're familiar with. – gunes May 07 '20 at 14:51
Which programming language did you use? – Ad van der Ven May 07 '20 at 15:34
My answer is pseudo-code. Ben has written it in R. – gunes May 07 '20 at 15:36
1

Now I understood. With p = 0.05, q = 0.03 and r = 0.01 in SPSS the syntax is: COMPUTE U=RV.UNIFORM(0,1). RECODE U (0.0 thru 0.03=0) (0.03 thru 0.94=1) (0.94 thru 0.99=2) (ELSE=3). EXECUTE. – Ad van der Ven May 08 '20 at 08:56
@AdvanderVen yes, that seems correct. – gunes May 08 '20 at 10:21
1

@ gunes Thanks a lot for all your help. – Ad van der Ven May 08 '20 at 14:48
@AdvanderVen if you find the answers useful, can you please accept and/or upvote it/them? – gunes May 08 '20 at 15:55
I do not know how to do this. – Ad van der Ven May 09 '20 at 08:04
@AdvanderVen to accept it, you’ll click the tick under the arrow next to the answer you want to accept. And if you want to upvote it as well, you’ll click the arrow pointing towards upwards which you can do it for any answer. – gunes May 09 '20 at 08:07

Simulating random variables from a discrete distribution II

2 Answers2