One way to obtain random integers from a known probability function is to
write integers on slips of paper and place them in a hat.
Draw one slip at a time.
After each drawing replace the slip in the hat.
The probability of
drawing the integer *i* is given by the ratio *a*_{i} of the number of slips
containing the integer *i* divided by the total number of slips.
Obviously the sum over *i* of *a*_{i} must be unity.
Another way to get random integers
is to throw one of a pair of dice.
Then all *a*_{i} equal zero except
.The probability that the
integer *i* will occur on the first drawing and the integer *j* will
occur on the second drawing is *a*_{i} *a*_{j}.
If you draw two slips or throw a pair of dice,
then the probability that the sum of *i* and *j* equals *k* is
readily seen to be

(53) |

(54) |

(55) |

(56) |

Consider the size of *A*(*Z*) for real .If , the sum of the terms of *A*(*Z*)
may be visualized in the complex plane as a sum of vectors
all pointing in the positive real direction.
If the vectors point in different directions.
This is shown in Figure 9.

4-8
The complex numbers
added together.
Figure 9 |

In raising to the *n*th power,
the values of of greatest concern are those near where *A* is
largest--because in any region where *A* is small
*A*^{n} will be extremely small.
Near or *Z* = 1 we may expand *A*(*Z*)
in a power series in

(57) |

(58) | ||

(59) | ||

(60) | ||

(61) | ||

(62) |

(63) | ||

(64) |

(65) | ||

(66) |

(67) | ||

(68) |

(69) |

- 1.
- The probability of drawing the number is one-half.
- 2.
- The probability of is one-half.
- 3.
- The probability of any other number is zero.

Now, raising to the *n*th power gives
a series in powers of whose coefficients are symmetrically
distributed about *Z* to the zero power and whose magnitudes are given
by the binomial coefficients. A sketch of the coefficients of *B*(*Z*)^{n}
is given in Figure 10.

Figure 10

We will now see how, for large *n*, the binomial coefficients
asymptotically approach a gaussian.
Approaching this limit is a bit tricky.
Obviously, the sum of *n* random integers will diverge as .Likewise the coefficients of powers of *Z* in
individually get smaller while the number of coefficients gets larger.
We recall that in time series analysis
we used th substitution .We commonly chose ,which had the meaning
that data points were given at integral points on the time axis.
In the
present probability theory application of *Z* transforms,
the choice
arises from our original statement that the numbers chosen
randomly from the slips of paper were integers.
Now we wish to add *n*
of these random numbers together;
and so, it makes sense to rescale the
integers to be integers divided by .Then we can make the
substitution .The coefficient of *Z*^{k} now refers to the probability of drawing the
number .Raising to the
*n*th power to find the probability distribution for the sum of *n*

(70) |

The probability that the number *t* will result from the sum is now found
by inverse Fourier transformation of (70).
The Fourier transform of
the gaussian (70) may be looked up in a table of integrals.
It is found to be the gaussian

10/30/1997