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## 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:

 X Prob(X) \$0 .5 \$1.00 .5

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):

 X-µ (X-µ )2 (X-µ )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.

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