Generating random numbers with known mean and variance

Question

From a paper I'm reading right know:

...
S(t+1, k) = S(t, k) + ... + C*∆
...
∆ is a standard random variable with mean 0 and variance 1.
...

How to generate this series of random values with this mean and variance? If someone has links to a C or C++ library I would be glad but I wouldn't mind implementing it myself if someone tells me how to do it :)

Answer 1

Do you have any restrictions on the distribution of \Delta ? if not you can just use a uniform distribution in [-sqrt(3), sqrt(3)]. The reason why this would work is because for an uniform distribution [a,b] the variance is 1/(12) (b-a)^2.

Answer 2

You can use the Box-Muller transform.

Suppose U₁ and U₂ are independent random variables that are uniformly distributed in the interval (0, 1]. Let

and

Then Z₀ and Z₁ are independent random variables with a normal distribution of standard deviation 1.