Math.random（）* 50 + Math.random（）* 20的分布与Math.random（）* 70的分布相比如何？

Question

How does the distribution of:

var randomNumber = Math.random()*50 + Math.random()*20;

compare to that of:

var randomNumber = Math.random()*70;

Answer 1

The first will not produce a flat distribution with more values near 70/2, while the second will produce an even distribution..

The easy way to find out is just to sample the values and graph them.

Sampled slowly just for fun.

 const ctx = canvas.getContext("2d"); const a1 = new Float64Array(70); const a2 = new Float64Array(70); var total = 0; function doSamples(samples){ for(var i = 0; i < samples; i ++){ var n1 = Math.random() * 50 + Math.random() * 20; var n2 = Math.random() * 70; a1[n1 | 0] += 1; a2[n2 | 0] += 1; } var max = 0; for(i = 0; i < 70; i ++){ max = Math.max(max,a1[i],a2[i]); } ctx.clearRect(0,0,canvas.width,canvas.height); for(i = 0; i < 70; i ++){ var l1 = (a1[i] / max) * canvas.height; var l2 = (a2[i] / max) * canvas.height; ctx.fillStyle = "Blue"; ctx.fillRect(i * 8,canvas.height - l1,4,l1) ctx.fillStyle = "Orange"; ctx.fillRect(i * 8 + 4,canvas.height - l2,4,l2) } total += samples; count.textContent = total; } function doit(){ doSamples(500); setTimeout(doit,100); } doit(); 

 canvas {border:2px solid black;} 

 <canvas id="canvas" width = 560 height = 200></canvas><br> Orange is random() * 70<br> Blue is random() * 50 + random() * 20<br> Graph is normalised. <span id="count"></span> samples. 

Answer 2

You could do a brute force approach by counting a million random values and check if the sum r70s is equal to a single random value r70 .

As you see the distribution is not equal.

 function countValue(key, value) { value = Math.floor(value); count[key][value] = (count[key][value] || 0) + 1; } var i, r20, r50, r70, count = { r20: [], r50: [], r70: [], r70s: [] }; for (i = 0; i < 1e6; i++) { r20 = Math.random() * 20; r50 = Math.random() * 50; r70 = Math.random() * 70; countValue('r20', r20); countValue('r50', r50); countValue('r70', r70); countValue('r70s', r20 + r50); } console.log(count); 

 .as-console-wrapper { max-height: 100% !important; top: 0; } 

Answer 3

The density function of a sum of random variables is the convolution of the density functions of the summands.

In this case the two summands have uniform densities, therefore their convolution is a piecewise linear function (a triangle). In general for the sum of n uniform variables, the density of the sum is a piecewise polynomial of degree n - 1.

The sum has the expected value equal to the sum of the expected values, namely 50/2 and 20/2, which is equal to 70/2, which is the expected value of Math.random()*70. So the expected values are the same, but the distributions are different.