Two variates A and B are statistically independent if the conditional probability (probability of an event A assuming that B has occurred) of A given B satisfies

(A.15) |

in which case the probability of A and B is just

(A.16) |

Similarly, n events are independent if

(A.17) |

Then the normal form variate

(A.18) |

has a limiting cumulative distribution function which approaches a normal distribution.

Under additional conditions on the distribution of the variates, the probability density itself is also normal with mean and variance . If conversion to normal form is not performed, then the variate

(A.19) |

is normally distributed with and .

(A.20) |

Now write

(A.21) | |

(A.22) |

so we have

(A.23) |

Now expand

(A.24) |

so

(A.25) |

since

(A.26) | |

(A.27) |

Taking the FOURIER transform

(A.28) |

This is of the form

(A.29) |

where and . This integral yields

(A.30) |

Therefore

(A.31) |

But and , so

(A.32) |

The ``fuzzy'' central limit theorem says that data which are influenced by many small and unrelated random effects are approximately normally distributed.

R. Minixhofer: Integrating Technology Simulation into the Semiconductor Manufacturing Environment