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Unformatted text preview: Stat 350 Gunderson Lecture Notes Chapter 8: Random Variables All models are wrong; some models are useful. -- George Box Patterns make life easier to understand and decisions easier to make. In Chapter 2 we discussed the different types of data or variables and how to turn the data into useful information with graphs and numerical summaries. Having some notion of probability from the previous chapter, we can now view the variables as “random variables” – the numerical outcomes of a random circumstance. We will look at the pattern of the distribution of the values of a random variable and we will see how to use the pattern to find probabilities. These patterns will serve as models in our inference methods to come. 8.1 What is a Random Variable? Recall in our discussion on probability we started out with some random circumstance or experiment that gave rise to our set of all possible outcomes S . We developed some rules for calculating probabilities about various events. Often the events can be expressed in terms of a “random variable” taking on certain outcomes. Loosely, this random variable will represent the value of the variable or characteristic of interest, but before we look . Before we look, the value of the variable is not known and could be any of the possible values with various probabilities, hence the name of a “random” variable. Definition: A random variable assigns a number to each outcome of a random circumstance, or, equivalently, a random variable assigns a number to each unit in a population. We will consider two broad classes of random variables: discrete random variables and continuous random variables. Definitions: A discrete random variable can take one of a countable list of distinct values. A continuous random variable can take any value in an interval or collection of intervals. 47 Try It! Discrete or Continuous A car is selected at random from a used car dealership lot. For each of the following characteristics of the car, decide whether the characteristic is a continuous or a discrete random variable. a. Weight of the car (in pounds). Continuous b. Number of seats (maximum passenger capacity). Discrete c . F i n i s h c o l o r . Discrete d. Overall condition of car (1 = good, 2 = very good, 3 = excellent). Discrete e. Length of car (in feet). Continuous In statistics, we are interested in the distribution of a random variable and we will use the distribution to compute various probabilities. The probabilities we compute (for example, p-values in testing theories) will help us make reasonable decisions. So just what is the distribution of a random variable? Loosely, it is a model that shows us what values are possible for that particular random variable and how often those values are expected to occur (i.e. their probabilities). The model can be expressed as a function or table or picture, depending on the type of variable it is....
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- Winter '08
- Normal Distribution, Probability distribution, Probability theory, probability density function, INDEP