Just a clarification

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 Random() is usually uniformly distributed, Random() * Random() is not.

Example

This is a uniform random distribution sample simulated through a pseudo-random variable:

``````        BarChart[BinCounts[RandomReal[{0, 1}, 50000], 0.01]]
``````

While this is the distribution you get after multiplying two random variables:

``````        BarChart[BinCounts[Table[RandomReal[{0, 1}, 50000] *
RandomReal[{0, 1}, 50000], {50000}], 0.01]]
``````

So, both are “random”, but their distribution is very different.

Another example

While 2 * Random() is uniformly distributed:

``````        BarChart[BinCounts[2 * RandomReal[{0, 1}, 50000], 0.01]]
``````

Random() + Random() is not!

``````        BarChart[BinCounts[Table[RandomReal[{0, 1}, 50000] +
RandomReal[{0, 1}, 50000], {50000}], 0.01]]
``````

The Central Limit Theorem

The Central Limit Theorem states that the sum of Random() tends to a normal distribution as terms increase.

With just four terms you get:

``````BarChart[BinCounts[Table[RandomReal[{0, 1}, 50000] + RandomReal[{0, 1}, 50000] +
Table[RandomReal[{0, 1}, 50000] + RandomReal[{0, 1}, 50000],
{50000}],
0.01]]
``````

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:

Edit

A few credits

Thanks to Thomas Ahle for pointing out in the comments that the probability distributions shown in the last two images are known as the Irwin-Hall distribution

Thanks to Heike for her wonderful torn[] function