The Central Limit Theorem (CLT)

The Central Limit Theorem (CLT) is a fundamental theorem in statistics that describes the behavior of the mean of a large number of independent, identically distributed (i.i.d.) random variables. In essence, the CLT states that the distribution of the sample mean approaches a normal distribution as the sample size increases, regardless of the original distribution of the data.

For a uniformly distributed variable \( U(a, b) \), the expected mean value \( \mu \) for a sample size \( n \) is calculated as:

\( \mu = \frac{a + b}{2} \)

Here, \( a = 0 \) and \( b = 10 \), so the expected mean is 5. The mean for each sample will approximate 5 as \( n \) increases.

Box-Muller transform from uniform to normal random series

Exponential variate from uniform variate

Exponential variate from uniform variate

uniform to normal distribution using central limit theorem