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copied!<h1>Just a clarification</h1> <p>Although the previous answers are right whenever you try to spot the randomness of a pseudo-random variable or its multiplication, you should be aware that while <strong>Random()</strong> is usually uniformly distributed, <strong>Random() * Random()</strong> is not. </p> <h2>Example</h2> <p>This is a <a href="http://en.wikipedia.org/wiki/Uniform_distribution_(continuous)" rel="noreferrer">uniform random distribution sample</a> simulated through a pseudo-random variable:</p> <p><img src="https://i.stack.imgur.com/M6XTU.png" alt="Histogram of Random()"> </p> <pre><code> BarChart[BinCounts[RandomReal[{0, 1}, 50000], 0.01]] </code></pre> <p>While this is the distribution you get after multiplying two random variables:</p> <p><img src="https://i.stack.imgur.com/S3Oaa.png" alt="Histogram of Random() * Random()"> </p> <pre><code> BarChart[BinCounts[Table[RandomReal[{0, 1}, 50000] * RandomReal[{0, 1}, 50000], {50000}], 0.01]] </code></pre> <p>So, both are “random”, but their distribution is very different.</p> <h2>Another example</h2> <p>While <strong>2 * Random()</strong> is uniformly distributed:</p> <p><img src="https://i.stack.imgur.com/XsFWW.png" alt="Histogram of 2 * Random()"></p> <pre><code> BarChart[BinCounts[2 * RandomReal[{0, 1}, 50000], 0.01]] </code></pre> <p><strong>Random() + Random() is not!</strong></p> <p><img src="https://i.stack.imgur.com/OyqWF.png" alt="Histogram of Random() + Random()"></p> <pre><code> BarChart[BinCounts[Table[RandomReal[{0, 1}, 50000] + RandomReal[{0, 1}, 50000], {50000}], 0.01]] </code></pre> <h2>The Central Limit Theorem</h2> <p><strong>The <a href="http://en.wikipedia.org/wiki/Central_limit_theorem" rel="noreferrer">Central Limit Theorem</a> states that the sum of <em>Random()</em> tends to a <a href="http://en.wikipedia.org/wiki/Normal_distribution" rel="noreferrer">normal distribution</a> as terms increase.</strong></p> <p>With just four terms you get:</p> <p><img src="https://i.stack.imgur.com/VWnJE.png" alt="Histogram of Random() + Random() + Random() + Random()"></p> <pre><code>BarChart[BinCounts[Table[RandomReal[{0, 1}, 50000] + RandomReal[{0, 1}, 50000] + Table[RandomReal[{0, 1}, 50000] + RandomReal[{0, 1}, 50000], {50000}], 0.01]] </code></pre> <p>And here you can see the road from a uniform to a normal distribution by adding up 1, 2, 4, 6, 10 and 20 uniformly distributed random variables:</p> <p><img src="https://i.stack.imgur.com/Apmgz.png" alt="Histogram of different numbers of random variables added"></p> <p><strong>Edit</strong></p> <p>A few credits</p> <p>Thanks to <a href="https://stackoverflow.com/users/205521/thomas-ahle">Thomas Ahle</a> for pointing out in the comments that the probability distributions shown in the last two images are known as the <a href="http://en.wikipedia.org/wiki/Irwin%E2%80%93Hall_distribution" rel="noreferrer">Irwin-Hall distribution</a> </p> <p>Thanks to <a href="https://mathematica.stackexchange.com/users/46/heike">Heike</a> for her wonderful <a href="https://mathematica.stackexchange.com/a/4155/193">torn[] function</a></p>

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