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## Discrete Probabilities and Probability Distributions

The simplest form of probability functions are discrete probabilities. In this case, we have some variable which can take different values or outcomes, .

Examples:
if represents a statement such as it will rain tomorrow'', it can take two values TRUE and FALSE; if represents the result of a coin flip, there would be the two possible outcomes, meaning heads and meaning tails; if is the number drawn in the next lottery draw, there would be 13,983,816 possibilities.
We denote the probability that takes a particular value as . The probability must be between and . If , takes the value with certainty; if , definitely does not take the value ; the closer is to , the greater the likelihood that takes this value. Assuming that can take only one of the values at a time (the values are mutually exclusive), the probability that takes or is . Since must take some value, the sum of the the probabilities of all possibly values is . Thus, the probability is normalised,

which means that always takes one of its allowed values.

We are often interested, however, in cases in which a random variable takes a continuous range of values. In this case, the above must be modified. We write a probability distribution or density as a function of the continuous variable .

Example:
A farmer might be interested in the probability that a given amount of rainfall will occur in a particular growing season. The rainfall, , could be modelled by a continuous positive real number .
We cannot interpret as the probability that takes the value , because that probability is almost always going to be . You cannot have an infinite number of possibilities with nonzero probability, because it could not be normalised. For continuous distributions, is defined so that the probability that is between and is , where is arbitrarily small. The normalisation condition converts from a sum to an integral, and is,

Next: Combining Probabilities Up: Basic Ideas Previous: Basic Ideas
Jon Shapiro
1999-09-23