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 we draw two slips or throw a pair of dice,
then the probability that
the sum of i and j equals k is
the sum of all the possible ways this can happen:
(57)
Since this equation is a convolution,
we may look into the meaning of the Z-transform
(58)
In terms of Z-transforms,
the probability that i plus j equals k is
simply the coefficient of Z^{k} in
(59)
The probability density of a sum
of random numbers is the convolution of their
probability density functions.
EXERCISES:
A random-number generator provides random integers 2, 3, and 6 with
probabilities
p(2)=1/2,
p(3)=1/3, and
p(6)=1/6.
What is the probability that any given integer n
is the sum of three of these random numbers?
(HINT: Leave the result in the form of coefficients
of a complicated polynomial.)