Chapter 4- Continuous.pdf - Probability Density Functions Cumulative DF and Expected Values Normal Distribution Exponential Distribution Gamma

Chapter 4- Continuous.pdf - Probability Density Functions...

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Probability Density Functions Cumulative DF and Expected Values Normal Distribution Exponential Distribution Gamma Distribution Chi-Square distribution Probability Plot Chapter 4: Continuous RV and Probability Distributions March 1, 2017 Chapter 4: Continuous RV and Probability Distributions
Probability Density Functions Cumulative DF and Expected Values Normal Distribution Exponential Distribution Gamma Distribution Chi-Square distribution Probability Plot Probability Density Functions Cumulative DF and Expected Values Normal Distribution Exponential Distribution Gamma Distribution Chi-Square distribution Probability Plot Chapter 4: Continuous RV and Probability Distributions
Probability Density Functions Cumulative DF and Expected Values Normal Distribution Exponential Distribution Gamma Distribution Chi-Square distribution Probability Plot Probability Density Functions Chapter 4: Continuous RV and Probability Distributions
Probability Density Functions Cumulative DF and Expected Values Normal Distribution Exponential Distribution Gamma Distribution Chi-Square distribution Probability Plot A discrete random variable (rv) is one whose possible values either constitute a finite set or else can be listed in an infinite sequence (a list in which there is a first element, a second element, etc.). A random variable whose set of possible values is an entire interval of numbers is not discrete. Definition Let X be a continuous rv. Then a probability distribution or probability density function (pdf) of X is a function f ( x ) such that for any two numbers a and b with a b . P ( a X b ) = R b a f ( x ) dx Chapter 4: Continuous RV and Probability Distributions
Probability Density Functions Cumulative DF and Expected Values Normal Distribution Exponential Distribution Gamma Distribution Chi-Square distribution Probability Plot For f ( x ) to be a legitimate pdf, it must satisfy the following two conditions: 1. f ( x ) 0 for all x 2. Z 1 1 f ( x ) dx = area under the entire graph of f ( x ) = 1 Chapter 4: Continuous RV and Probability Distributions
Probability Density Functions Cumulative DF and Expected Values Normal Distribution Exponential Distribution Gamma Distribution Chi-Square distribution Probability Plot Example 4.4 The direction of an imperfection with respect to a reference line on a circular object such as a tire, brake rotor, or flywheel is, in general, subject to uncertainty. Consider the reference line connecting the valve stem on a tire to the center point, and let X be the angle measured clockwise to the location of an imperfection. One possible pdf for X is f ( x ) = ( 1 360 0 x < 360 0 otherwise Clearly f ( x ) 0. The area under the curve is just the area of the rectangle (height)(base)= 1 360 (360) = 1 The probability hat the angle is between 90 o and 180 o is P (90 X 180) = Z 180 90 1 360 dx = x 360 | x =180 x =90 = 1 4 = 0 . 25 Chapter 4: Continuous RV and Probability Distributions
Probability Density Functions Cumulative DF and Expected Values Normal Distribution Exponential Distribution Gamma Distribution Chi-Square distribution Probability Plot Definition