# 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