08-Continuous RVs

08-Continuous RVs - IE 111 Fall Semester 2009 Lecture Note...

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IE 111 Fall Semester 2009 Lecture Note Set 8 Continuous Random Variables and Distributions Continuous random variables have a domain defined on the set of real numbers. Typically the domain is an interval (e.g. all real numbers between [a,b]), a ray (e.g. all real numbers greater than c), or the whole set of real numbers. Some examples The height or weight of a randomly selected person. The time between now and when the next bus arrives. The measured length of some part. The error in the measured length of some part. The GPA of a randomly selected student. The percent sugar in a randomly selected can of soda. Associated with continuous random variables is a "probability density function". This serves the same purpose as the probability mass function for discrete random variables. It shows how likely various events and outcomes are. Because we are dealing with real numbers instead of integers, we must be careful how we define things. The first property of continuous random variables is: P(X=x) = 0 for all x This is due to the real numbered domain. Let X be the height of a randomly selected person. What is the probability that that person is 56.666666666666666 inches tall? Remember that since we are dealing with real numbers, the height must be accurate to an infinite number of decimal places! Note that for a continuous random variable: P(X b) = P(X<b) + P(X=b) = P(X<b) Hopefully this illustrates the problem with real numbers. To get around this problem we define the PDF as follows: 1 = < < b a X dx x f b X a P ) ( ) (
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The PDF also has the following two properties f X (x) 0 Note that it is not necessarily true that f X (x) 1. Neither is it true that f X (x) is a probability. Example Let X have the following continuous distribution: f X (x) = 10 for 1 X 1.1 Is this a valid PDF? Yes because it is true that the area under the curve integrates to 1. To make things easier to write, we will now let f(x) = f X (x) Cumulative Distribution Function As before F X (x) = P(X x) Which is the area under the curve over the range X x 2 1 ) ( = - dx x f X - = x du u f ) (
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Mean and Variance The following definitions apply: Now a couple of examples: Example 1. Consider the following distribution, A.K.A. probability density function or PDF: f X (x) = x for 0 x < 1 f X (x) = 2-x for 1 x 2 f X (x) = 0 Otherwise a) Verify that this is a valid density xdx 0 1 + 2 1 2 - xdx = [ x 2 /2 ] 0 1 + [ 2x - x 2 /2 ] 1 2 = [ (1/2)-(0) ] + [ (4 - 4/2) - (2 - 1/2) ] = (1/2) + (2 - 3/2) = 1/2 + 1/2 = 1 b) Find F X (x) For 0 < x < 1 we have x xdx 0 = [ x 2 /2 ] 0 x = x 2 /2 3 [ ] - - - - - = - = = = = dx x f X E x X V X E X E X V dx x f x g X g E dx x f x X E dx x xf X E ) ( )) ( ( ) ( )] ( [ ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( 2 2 2 2 2
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For 1 < x < 2 we have xdx 0 1 + 2 1 - xdx x = 1/2 + [ 2x - x 2 /2 ] 1 x = 1/2 + (2x - x 2 /2 ) - (2 - 1/2) = 2x - x 2 /2 - 1
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08-Continuous RVs - IE 111 Fall Semester 2009 Lecture Note...

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