How to calculate PMF of sample median for discrete random variable in R?

Question

A bag contains 5 billiard balls numbered 1, 2, 3, 4, 5. A random sample of size n = 3 is drawn without replacement from the bag. What is the probability mass function of the sample median?

Here is what I have:

library(listviewer) sampleSpaceAndMedian = list()

# the random samples (1,2,3), (1,3,2), (2,1,3), 
# (2,3,1), (3,1,2), and (3,2,1) have the same mean
# therefore, belong to the same equivalence class
for (a in 1:3){
  for (b in 2:4){
    for (c in 3:5){
      # a unique random sample of size 3 (ignores the order)
      if (b > a && c > b){
        tString = paste(toString(a), toString(b), toString(c), toString(median(c(a,b,c))), sep = " ")
        sampleSpaceAndMedian <- append(sampleSpaceAndMedian, tString)
      }
    }
  }    
}
# the random sample is in the first three columns
# median is the fourth column
jsonedit( sampleSpaceAndMedian )    
```

Can you please help me to get the PMF? Thanks.

Answer 1

You can use combn to get all the combinations of a vector and apply a function to it:

combn(1:5, 3)
#     [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10]
#[1,]    1    1    1    1    1    1    2    2    2     3
#[2,]    2    2    2    3    3    4    3    3    4     4
#[3,]    3    4    5    4    5    5    4    5    5     5

To get the distribution of the median you can use the following:

prop.table(table(combn(1:5, 3, median)))

#>  2   3   4 
#>0.3 0.4 0.3