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The concept of random variables and their classification as discrete or continuous. Discrete random variables have a finite or countable number of possible outcomes, while continuous random variables have an uncountable number of possible outcomes. The document also covers probability distributions for discrete random variables, which list each possible value and its probability, and provides examples of probability distributions and their expected value, variance, and standard deviation calculations.

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Download Understanding Discrete and Continuous Random Variables and Their Probability Distributions and more Study Guides, Projects, Research Probability and Statistics in PDF only on Docsity! Section 4.1, Probability Distributions A random variable x represents a numerical value associated with each outcome of a probability experiment. (Note: Many other books make a distinction between an unknown random variable X and the value that it takes x.) There are two kind of random variables: discrete and continuous. The kind of random variable determines how we will analyze it. A random variable is discrete if it has a finite number of possible outcomes, or a countable number (i.e. the integers are infinite, but are able to be counted). For example, the number of heads you get when flip a coin 100 times is discrete, since it can only be a whole number between 0 and 100. A random variable is continuous if it has an uncountable number of possible outcomes, such as an interval on a number line (the interval does not need to be bounded). For example, the amount of time you spend studying for this class is a continuous random variable, since it can be any number that is at least zero hours, and you can spend any percentage of an hour or minute studying. Examples Determine whether each of the following variables x is discrete or continuous: 1. Let x represent the number of times you do laundry this month. Discrete 2. Let x represent your annual salary given to the nearest cent. Discrete 3. Let x represent a car’s speed as it drives past the 3300 South exit on northbound I-15. Con- tinuous 4. Let x represent your height at age 10. Continuous 5. Let x represent number of math classes that you have taken in your life. Discrete We’ll be spending the rest of this chapter analyzing discrete random variables. In Chapter 5 we will be working with a very important class of continuous random variables called Normal random variables. 1 Discrete Probability Distributions A discrete probability distribution lists each possible value that a random variable can take, along with its probability. It has the following properties: • The probability of each value of the discrete random variable is between 0 and 1, so 0 ≤ P (x) ≤ 1. • The sum of all the probabilities is 1, so ∑ P (x) = 1. Examples Determine if each of the following tables represents a probability distribution: 1. x −5 6 9 P (x) 0.5 0.25 0.25 Yes, this is a probability distribution, since all of the probabilities are between 0 and 1, and they add to 1. 2. x 1 2 3 4 P (x) 0.4 0.4 0.4 0.2 This is not a probability distribution, since the probabilities add to 1.4, not 1. 1