Discrete Probability Distributions
A probability distribution is a rule that attaches
probabilities to possible outcomes. The set of possible
outcomes is called the sample space. In order for a rule to
be a valid probability distribution, it must never have a
probability outside the range of zero to one, and the sum of
probabilities over the entire sample space must be one.
There are two kinds of sample spaces, discrete and
continuous. The possible outcomes in a discrete sample space
can be counted (thought they could be infinite), while the
possible outcomes in a continuous sample space are
measured.
A simple example of a discrete probability distribution
is a coin flip with a payoff. Suppose that you win $1 when
heads appears and win nothing when tails shows. The
probability distribution that describes this sample
situation is a sample space of 0, $1, each with a
probability of 1/2, or:
A probability distribution has a mean, often called the
expected value. It represents what we expect the
average to be if we could run the process that the
distribution describes a great many times. In our example
here, if we flip a coin a great many times, we expect half
the time to get zero and half the time to win $1.00. Hence,
the expect value is .5($0) + .5($1.00) or $.50.
Probability distributions can also have standard
deviations. For a discrete probability distribution, we
subtract the mean or expect value from possible outcome,
square the result, multiply it by the probability of the
outcome, sum the results, and finally take the square root
of the sum. In the case of our simple distribution above,
the computation is shown below (without the dollar
signs):
Xµ

(Xµ )^{2}

(Xµ )^{2}(Prob(X))

0.5 = .5

.25

.25*.5 =. 125

1.00.5 = .5

.25

.25*.5 = .125

Adding the results gives .25. Taking the square root
yields .5. If we consider the standard deviation as the
average distance we expect to be away from the mean, this
result is not surprising. In this case we will always be .5
away from the expected value.
