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LectureNotes7 - Continuous Random Variable and Their...

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Continuous Random Variable and Their Probability Distributions: Part I Cyr Emile M’LAN, Ph.D. [email protected] Continuous Random Variables: Part I – p. 1/34
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Introduction Text Reference : Introduction to Probability and Its Application, Chapter 5. Reading Assignment : Sections 5.1-5.2, 5.9, 5.10, March 16 - March 18 As mentioned before, there are two types of random variables: discrete and continuous. In this set of notes, we study the second types of random variable that arises in many applied problems. Continuous Random Variables: Part I – p. 2/34
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Continuous Random Variables A random variable is continuous if the values it can assumed cannot be enumerated (can be represented by an interval). Example 5.1 : If a chemical compound is randomly selected and its pH X is determine, then X is a continuous random variable because any pH value between 0 and 14 is possible. If in the study of the ecology of a lake, we make depth measurements at randomly chosen locations, then X = the depth at such location is a continuous random variable. Continuous Random Variables: Part I – p. 3/34
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Continuous Random Variables Unlike discrete random variables, it is impossible to assign nonzero probabilities to all the possible values of a continuous random variable. Question: How do deal with this type of random variable? The idea is to find an alternative representation to the probability mass function that will characterize continuous distributions. For, "discretize" the random variable. The resulting discrete distribution can be pictured using a probability histogram. As one refines the discretization, a much smoother curve appears on top of the histogram plot. If we continue in this way, the sequence of histograms approaches a smooth curve. Continuous Random Variables: Part I – p. 4/34
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Continuous Random Variables Histogram of x x Density -4 -2 0 2 4 0.0 0.1 0.2 0.3 0.4 Histogram of x x Density -4 -2 0 2 4 0.0 0.1 0.2 0.3 0.4 Histogram of x x Density -4 -2 0 2 4 0.0 0.1 0.2 0.3 Histogram of x x Density -4 -2 0 2 4 0.0 0.1 0.2 0.3 0.4 Continuous Random Variables: Part I – p. 5/34
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Continuous Random Variables In any of these histograms, the area of all rectangles is 1. Thus, the total area under the smooth curve should also be 1. The probability that the random variable, X , falls between two values a and b , P ( a X b ) , is approximately equal to the sum of all rectangles that fall in the interval [ a, b ] . Thus, the total area under the smooth curve between a and b is just the probability that the random variable falls in the interval [ a, b ] . The smooth curve is called a probability density function and it is the quantity that is used to characterize continuous distribution. Continuous Random Variables: Part I – p. 6/34
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Continuous Random Variables Definition 5.1 : Let X be a continuous random variable. Then there exists a mathematical function f ( x ) called the the probability density function of the random variable. X satisfying 1. f ( x ) 0 for all -∞ < x < .
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