How to create a vector where indices of non-zero elements follow a distribution

I need to write a program that will create a vector of size N that will contain K non-zero elements according to the following requirements:

Non-zero elements should be mostly concentrated near the middle element (at position N/2) of the vector.
Elements at distance D or further from the middle element (on either side) should be zero.
As we move away from the middle element, the probability that an element is non-zero should be decreasing.

A rather small example of what I would like to accomplish follows, where N = 40 (middle element is 20), K = 11 non-zero elements, and D = 8. Since D = 8, elements at positions > 20 + 8 = 28 and elements at positions < 20 - 8 = 12 should always be zero. In the zone where non-zeros are allowed (positions from 12 to 28) K = 11 non-zero elements are present. There are more non-zero elements close to position 20 and they become more sparse as we move further away from the middle element.

Position	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20	21	22	23	24	25	26	27	28	29	30	31	32	33	34	35	36	37	38	39	40
Vector	0	0	0	0	0	0	0	0	0	0	0	0	1	0	1	1	0	1	1	1	1	1	0	1	1	0	0	1	0	0	0	0	0	0	0	0	0	0	0	0

I have not yet written any code, since I cannot wrap my head around on how to even start. One idea I had was to somehow use the binomial distribution to generate random indices and set the non-zero elements. However, this distribution can give multiple times the same index and hence less than K non-zero elements will be produced. If I use a loop to generate new random numbers until a non-used index is found, will the result still follow a binomial distribution, so that more non-zero elements will be around the middle element?

The programming language that will be used is not that important, but I would prefer something in Matlab, Python, C++ or C, as I am more familiar with them.

I hope someone can provide directions and/or examples.

Solution

This is existing functionality in numpy (choice)

import numpy as np
from scipy import stats

N = 40
K = 11

Your vague description of the distribution you want is not adequate, so I'm just going to use a normal probability distribution with a mean of N/2 and a standard deviation of sqrt(N/2).

center = int(N / 2)
scale = np.sqrt(N / 2)

Create a probability vector from the probability density function for each possible index (up to N):

p = stats.norm(loc=center, scale=scale).pdf(np.arange(N))

Make sure it sums to 1:

p /= np.sum(p)

Initialize a random number generator and call .choice() on the possible indices, with the probability distribution p, setting replace to False:

rng = np.random.default_rng()
nz_indices = rng.choice(np.arange(N), size=K, p=p, replace=False)

>>> nz_indices
array([27, 20, 23, 19, 16, 24, 13, 25, 26, 22, 21])