Java: random integer with non-uniform distribution

Question

How can I create a random integer n in Java, between 1 and k with a "linear descending distribution", i.e. 1 is

Answer 1

So we need the following distribution, from least likely to most likely:

*
**
***
****
*****

etc.

Lets try mapping a uniformly distributed integer random variable to that distribution:

1
2  3
4  5  6
7  8  9  10
11 12 13 14 15

etc.

This way, if we generate a uniformly distributed random integer from 1 to, say, 15 in this case for K = 5, we just need to figure out which bucket it fits it. The tricky part is how to do this.

Note that the numbers on the right are the triangular numbers! This means that for randomly-generated X from 1 to T_n, we just need to find N such that T_(n-1) < X <= T_n. Fortunately there is a well-defined formula to find the 'triangular root' of a given number, which we can use as the core of our mapping from uniform distribution to bucket:

// Assume k is given, via parameter or otherwise
int k;

// Assume also that r has already been initialized as a valid Random instance
Random r = new Random();

// First, generate a number from 1 to T_k
int triangularK = k * (k + 1) / 2;

int x = r.nextInt(triangularK) + 1;

// Next, figure out which bucket x fits into, bounded by
// triangular numbers by taking the triangular root    
// We're dealing strictly with positive integers, so we can
// safely ignore the - part of the +/- in the triangular root equation
double triangularRoot = (Math.sqrt(8 * x + 1) - 1) / 2;

int bucket = (int) Math.ceil(triangularRoot);

// Buckets start at 1 as the least likely; we want k to be the least likely
int n = k - bucket + 1;

n should now have the specified distribution.