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Probability Distributions

# Probability Distributions - A PRIMER FOR RANDOM VARIABLES...

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A PRIMER FOR RANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS by Dr. Harvey A. Singer 1 Introduction A random variable is a variable that takes on numerical values determined by the outcome of a random experiment. A random variable is discrete if it can take on no more than a countable number of values. Discrete random variables generally result from a counting process. A random variable is continuous if it can take any value in an interval. Continuous random variables generally result from a measuring process. The two obvious questions regarding random variables are: (1) What is typical or expected of the random variable? (2) What are the probabilities that particular values or ranges of values will occur? The concept of expected value addresses questions of the first type, and the concept of probability distributions addresses questions of the second type. A probability distribution for a random variable, whether discrete or continuous, may generally be thought of as a theoretical listing of outcomes and their probabilities, which often may be obtained from a mathematical model representing the phenomenon of interest. A model is idealized representation of some underlying phenomenon. In particular, a mathematical model is a mathematical expression representing some underlying phenomenon. For discrete random variables, this mathematical expression is known as a probability distribution function . A probability distribution for a discrete random variable is simply a listing of the probabilities of all possible outcomes, according to the values of the random variable. For continuous random variables, the mathematical model representing some underlying phenomenon is known as a probability density function . When such a mathematical expression is available, the probability that various values of the random variable occur within specified ranges or intervals may be calculated. © 1999 by Ha rvey A. Singer 1

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(However the exact probability of a particular value is zero.) 2 Probability Distributions for Discrete Random Variables 2.1 Definition A probability distribution (or probability function or probability distribution function) for a discrete random variable is a mutually exclusive listing of all possible outcomes for that random variable such that a particular probability of occurrence is associated with each occurrence. In particular, the probability distribution of a discrete random variable X expresses the probability that X takes on the particular value x , as a function of x . That is, P X ( x ) = P ( X = x ) 2.2 Properties of Probability Functions of Discrete Random Variables 1. P X ( x ) ³ 0 for any value x of X . 2. The individual probabilities sum to 1, that is, P X x ( 29 = 1 x where the summation is over all possible values x of X . 2.3 Cumulative Probability Functions
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Probability Distributions - A PRIMER FOR RANDOM VARIABLES...

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