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L12posted

# L12posted - Continuous Random Variables Continuous random...

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Continuous Random Variables Continuous random variables are random variables that arise from measurements can take any value in interval Example .Random numbers in [0 , 1] used in random number genera- tors. 1

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Think of a circle with circumfer- ence 1 and a pointer attached in the center. Give it a long spin. Points to a “random” direction. 0.5 0.75 0.25 0 X =location of the pointer. Note . For any point a, we have P ( X = a ) = 0 2

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Described by a probability density function f ( x ) 0 f ( x ) < Total area under the graph is 1 Z + -∞ f ( x ) dx = 1 Interpretation. P ( a < X b ) = R b a f ( x ) dx Graphical Meaning. 3

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Uniform f ( x ) = 1 b - a , if a < x < b 0 , otherwise 4
Triangular 5

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Exponential f ( x ) = 1 μ e - x μ if x 0 0 otherwise 6
Loss of Memory Property of Ex- ponential . Let X be exponential and a, b > 0 P ( X > b + a | X > a ) = P ( X > b + a, X > a ) P ( X > a ) = P ( X > b + a ) P ( X > a ) = e - - ( b + a ) μ e - a μ = e - a μ = P ( X > b ) 7

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Normal (also called Gaussian) f ( x ) = 1 σ 2 π e - 1 2 x - μ σ 2 8

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Expected Value Definition . Expected value of
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