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Unformatted text preview: Chapter 4 Continuous random variables and Probability Distribution STA 113 Spring 2008 Jayanta Kumar Pal 1 / 36 Chapter 4 Outline 1 Introduction 2 Cumulative Distribution Functions and Expected Values 3 Normal Distributions 4 Other Continuous Distributions 5 Probability Plots 2 / 36 Chapter 4 Introduction Outline 1 Introduction 2 Cumulative Distribution Functions and Expected Values 3 Normal Distributions 4 Other Continuous Distributions 5 Probability Plots 3 / 36 Chapter 4 Introduction Random variables with candidate values neither finite nor countably infinite. Instead, an entire interval or a collection of such intervals form the range of the RV. Further, P ( X = c ) = 0 for any c , which is a possible value of X , the RV e.g. pH of soil measurement. Altitude from the sealevel of all places in the United States. Depth of a lake at any location. 4 / 36 Chapter 4 Introduction Sometimes the RV can be a mixture of discrete and continuous. e.g. X = waiting time for a customer in a parlor to get a haircut. Here P ( X = ) > 0, since the customer need not wait if the parlor is not fully occupied. Given X > 0, it is a continuous RV. Though some variables are intrinsically continuous, the limitations of measurement can render it discrete. e.g. height, weight, temperature. The models should be treated as continuous in those cases, since the person whose weight is reported as 154.5 lb, weighs anything between 154.45 and 154.55 lbs in reality. 5 / 36 Chapter 4 Introduction Probability density function Since P ( X = x ) = 0 for all x ; p ( x ) , the pmf does not make sense. Histogram idea is extended to finer and finer grids. For a continuous RV X , the probability density function (pdf) of X is defined as a function f ( x ) such that, P ( a X b ) = Z b a f ( x ) dx for any a b . f ( x ) 0 for all x . R  f ( x ) dx = 1. The graph of f ( x ) density curve. 6 / 36 Chapter 4 Introduction Uniform distribution A continuous RV X has a uniform distribution on the interval [ A , B ] if the pdf of X is f ( x ; A , B ) = 1 B A , A x B and 0 otherwise. The density curve is a flat function. For any A A < B B , we have P ( A X B ) = B A B A 7 / 36 Chapter 4 Introduction Let X be a continuous RV with density f ( x ) . Clearly, for any c , P ( X = c ) = lim P ( c X c + ) = lim R c + c f ( x ) dx = 0. Therefore, for any a b , P ( a < X < b ) = P ( a X < b ) = P ( a < X b ) = P ( a X b ) = Z b a f ( x ) area under the density curve between a and b . In essence, for continuous RV, it does not matter whether the probabilities are computed for open intervals or closed intervals. 8 / 36 Chapter 4 Introduction Example 4.5 X = time headway between two consecutive cars on a freeway....
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 Spring '08
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