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80 HANDBOOK OF PORTFOLIO MATHEMATICS
Suppose we are discussing tossing a single die. If we are talking about
having the outcome of 5, how many times will we have to toss the die, on
average, to achieve this outcome? The mean of the Geometric Distribution
tells us this. If we know the probability of throwing a 5 is 1/6 (.1667), then
the mean is 1/.1667=6. Thus, we would expect, on average, to toss a die
six times in order to get a 5. If we kept repeating this process and recorded
how many tosses it took until a 5 appeared, plotting these results would
yield the Geometric Distribution function formulated in (2.39).The Hypergeometric Distribution
Another type of discrete distribution related to the preceding distributions
is termed theHypergeometric Distribution. Recall that in the Binomial
Distribution it is assumed that each draw in succession from the popula-
tion has the same probabilities. That is, suppose we have a deck of 52 cards;
26 of these cards are black and 26 are red. If we draw a card and record
whether it is black or red, we then put the card back into the deck for the
next draw. This “sampling with replacement” is what the Binomial Distri-
bution assumes. Now, for the next draw, there is still a .5 (26/52) probability
of the next card’s being black (or red).
The Hypergeometric Distribution assumes almost the same thing, ex-
cept there is no replacement after sampling. Suppose we draw the first card
and it is red, and wedo notreplace it back into the deck. Now, the proba-
bility of the next draw’s being red is reduced to 25/51 or .4901960784. In the
Hypergeometric Distribution there isdependency, in that the probabilities
of the next event are dependent on the outcome(s) of the prior event(s).
Contrast this to the Binomial Distribution, where an event isindependent
of the outcome(s) of the prior event(s).
The basic functions N′(X) and N′(X) of the Hypergeometric are the
same as those for the Binomial, (2.33) and (2.35), respectively, except that
with the Hypergeometric the variable P, the probability of success on a
single trial, changes from one trial to the next.
It is interesting to note the relationship between the Hypergeometric
and Binomial Distributions. As N becomes larger, the differences between
the computed probabilities of the Hypergeometric and the Binomial draw
closer to each other. Thus, we can state that as N approaches infinity, the
Hypergeometric approaches the Binomial as a limit.
If you want to use the Binomial probabilities as an approximation of
the Hypergeometric, as the Binomial is far easier to compute, how big
must the population be? It is not easy to state with any certainty, since the