Understanding "randomness"

Question

I can't get my head around this, which is more random?

rand() 

OR:

rand() * rand() 

I´m finding it a real brain teaser, could you help me out?

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EDIT:

Intuitively I know that the mathematical answer will be that they are equally random, but I can't help but think that if you "run the random number algorithm" twice when you multiply the two together you'll create something more random than just doing it once.

Answer 1

I guess both methods are as random although my gutfeel would say that rand() * rand() is less random because it would seed more zeroes. As soon as one rand() is 0, the total becomes 0

Answer 2

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