How to generate random numbers biased towards one value in a range?

Question

Say, if I wanted to generate an unbiased random number between min and max, I'd do:

var rand = function(min, max) {
    return Math.floor(Math.random() * (max - min + 1)) + min;
};

But what if I want to generate a random number between min and max but more biased towards a value N between min and max to a degree D? It's best to illustrate it with a probability curve:

Answer 1

Here is one way:

• Get a random number in the min-max range
• Get a random normalized mix value
• Mix random with bias based on random mix

Ie., in pseudo:

Variables:
  min = 0
  max = 100
  bias = 67      (N)
  influence = 1  (D) [0.0, 1.0]

Formula:
  rnd = random() x (max - min) + min
  mix = random() x influence
  value = rnd x (1 - mix) + bias x mix

The mix factor can be reduced with a secondary factor to set how much it should influence (ie. mix * factor where factor is [0, 1]).

Demo

This will plot a biased random range. The upper band has 1 as influence, the bottom 0.75 influence. Bias is here set to be at 2/3 position in the range. The bottom band is without (deliberate) bias for comparison.

var ctx = document.querySelector("canvas").getContext("2d");
ctx.fillStyle = "red"; ctx.fillRect(399,0,2,110);  // draw bias target
ctx.fillStyle = "rgba(0,0,0,0.07)";

function getRndBias(min, max, bias, influence) {
    var rnd = Math.random() * (max - min) + min,   // random in range
        mix = Math.random() * influence;           // random mixer
    return rnd * (1 - mix) + bias * mix;           // mix full range and bias
}

// plot biased result
(function loop() {
  for(var i = 0; i < 5; i++) {  // just sub-frames (speedier plot)
    ctx.fillRect( getRndBias(0, 600, 400, 1.00),  4, 2, 50);
    ctx.fillRect( getRndBias(0, 600, 400, 0.75), 55, 2, 50);
    ctx.fillRect( Math.random() * 600          ,115, 2, 35);
  }
  requestAnimationFrame(loop);
})();

<canvas width=600></canvas>

Answer 2

Fun: use the image as the density function. Sample random pixels until you get a black one, then take the x co-ordinate.

Code:

getPixels = require("get-pixels"); // npm install get-pixels

getPixels("distribution.png", function(err, pixels) {
  var height, r, s, width, x, y;
  if (err) {
    return;
  }
  width = pixels.shape[0];
  height = pixels.shape[1];
  while (pixels.get(x, y, 0) !== 0) {
    r = Math.random();
    s = Math.random();
    x = Math.floor(r * width);
    y = Math.floor(s * height);
  }
  return console.log(r);
});

Example output:

0.7892316638026386
0.8595335511490703
0.5459279934875667
0.9044852438382804
0.35129814594984055
0.5352215224411339
0.8271261665504426
0.4871773284394294
0.8202084102667868
0.39301465335302055

Scale to taste.

Answer 3

Just for fun, here's a version that relies on the Gaussian function, as mentioned in SpiderPig's comment to your question. The Gaussian function is applied to a random number between 1 and 100, where the height of the bell indicates how close the final value will be to N. I interpreted the degree D to mean how likely the final value is to be close to N, and so D corresponds to the width of the bell - the smaller D is, the less likely is the bias. Clearly, the example could be further calibrated.

(I copied Ken Fyrstenberg's canvas method to demonstrate the function.)

function randBias(min, max, N, D) {
  var a = 1,
      b = 50,
      c = D;

  var influence = Math.floor(Math.random() * (101)),
    x = Math.floor(Math.random() * (max - min + 1)) + min;

  return x > N 
         ? x + Math.floor(gauss(influence) * (N - x)) 
         : x - Math.floor(gauss(influence) * (x - N));

  function gauss(x) {
    return a * Math.exp(-(x - b) * (x - b) / (2 * c * c));
  }
}

var ctx = document.querySelector("canvas").getContext("2d");
ctx.fillStyle = "red";
ctx.fillRect(399, 0, 2, 110);
ctx.fillStyle = "rgba(0,0,0,0.07)";

(function loop() {
  for (var i = 0; i < 5; i++) {
    ctx.fillRect(randBias(0, 600, 400, 50), 4, 2, 50);
    ctx.fillRect(randBias(0, 600, 400, 10), 55, 2, 50);
    ctx.fillRect(Math.random() * 600, 115, 2, 35);
  }
  requestAnimationFrame(loop);
})();

<canvas width=600></canvas>

Answer 4

Say when you use Math.floor(Math.random() * (max - min + 1)) + min;, you are actually creating a Uniform distribution. To get the data distribution in your chart, what you need is a distribution with non-zero skewness.

There are different techniques to get those kinds of distributions. Here is an example of beta distribution found on stackoverflow.

---

Here is the example summarized from the link:

unif = Math.random()  // The original uniform distribution.

And we can transfer it into beta distribution by doing

beta = sin(unif*pi/2)^2 // The standard beta distribution

To get the skewness shown in your chart,

beta_right = (beta > 0.5) ? 2*beta-1 : 2*(1-beta)-1;

You can change the value 1 to any else to have it skew to other value.