# 3: Distributions, Probability, and Expected Values

- Page ID
- 151675

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- 3.1: The Distribution Function as a Summary of Experimental Results
- As we collect increasing amounts of data, the accumulation quickly becomes unwieldy unless we can reduce it to a mathematical model. We call the mathematical model we develop a distribution function, because it is a function that expresses what we are able to learn about the data source—the distribution. A distribution function is an equation that summarizes the results of many measurements; it is a mathematical model for a real-world source of data.

- 3.2: Outcomes, Events, and Probability
- We also need to introduce the idea that a function that successfully models the results of past experiments can be used to predict some of the characteristics of future results.

- 3.3: Some Important Properties of Events
- If we know the probabilities of the possible outcomes of a trial, we can calculate the probabilities for combinations of outcomes. These are based on two rules, which we call the laws of probability. If we partition the outcomes into exhaustive and mutually exclusive events, the laws of probability also apply. Since, as we define them, “events” is a more general term than “outcomes,” we call them the law of the probability of alternative events and the law of the probability of compound events.

- 3.4: Applying the Laws of Probability
- The laws of probability apply to events that are independent. If the result of one trial depends on the result of another trial, we may still be able to use the laws of probability. However, to do so, we must know the nature of the interdependence.

- 3.5: Bar Graphs and Histograms
- Since a discrete distribution is completely specified by the probabilities of each of its events, we can represent it by a bar graph. The probability of each event is represented by the height of one bar. We can generalize this graphical representation to represent continuous distributions. To see what we have in mind, let us consider a particular example.

- 3.6: Continuous Distribution Functions - the Envelope Function is the Derivative of the Area
- When we can represent the envelope curve as a continuous function, the envelope curve is the derivative of the cumulative probability distribution function: The cumulative distribution function is f(u) ; the envelope function is df(u)/du . The envelope function is a probability density, and we will refer to the envelope function, df(u)/du , as the probability density function. The probability density function is the derivative, with respect to the random variable, of the cumulative distributi

- 3.9: Random Variables, Expected Values, and Population Sets
- When we sample a particular distribution, the value that we obtain depends on chance and on the nature of the distribution described by the function f(u) . The probability that any given trial will produce u in the interval a<u<b is equal to f(b)−f(a).

- 3.10: Statistics - the Mean and the Variance of a Distribution
- There are two important statistics associated with any probability distribution, the mean of a distribution and the variance of a distribution.

- 3.12: The Normal Distribution
- The normal distribution is very important. The central limit theorem says that if we average enough values from any distribution, the distribution of the averages we calculate will be the normal distribution.