16.1 Cont Rand Var

16.1 Cont Rand Var - cumulative distribution function •...

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Click to edit Master subtitle style 1/5/10 Continuous Random Variables Week of March 16
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1/5/10 Intro Domain defined on set of real numbers Unlike discrete r.v.’s that are finite or countably infinite Domain will be interval: all real numbers between [a,b] Or a ray, all real numbers greater than c Or the entire set of reals: xH (-• , & )
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1/5/10 PDF Analogous to discrete case and the pmf we can define a probability density function Serves same purpose Shows how likely various events / outcomes are 1st Property: P(X= x ) = 0 for all x
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1/5/10 Properties of PDF 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: = < < b a X dx x f b X a P ) ( ) ( 1 ) ( = - dx x f X
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1/5/10 Example Let X have the following continuous distribution: fX(x) = 10 Is this a valid PDF?
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1/5/10 CDF As in the discrete case, we can define the
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Unformatted text preview: cumulative distribution function : • Which is the area under the curve over the range Xx ± ∫ ∞-= = ≤ x X du u f x F x X P ) ( ) ( ) ( 1/5/10 Mean and Variance • The following definitions apply: [ ] ∫ ∫ ∫ ∫ ∞ ∞-∞ ∞-∞ ∞-∞ ∞--=-= = = = dx x f X E x X V X E X E X V dx x f x X E dx x f x g X g E dx x xf X E ) ( )) ( ( ) ( )] ( [ ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( 2 2 2 2 2 1/5/10 Example 1 Consider the following distribution, A.K.A. probability density function or PDF: fX(x) = x for 0 & x < 1 fX(x) = 2-x for 1 & x E 2 fX(x) = 0 Otherwise a) Verify that this is a valid density 1/5/10 Example 1 b) Find FX(x) For 0 < x < 1 we have For 1 < x < 2 we have 1/5/10 Example 1 d) Find E(X) e) Find V(X) 1/5/10 Example 2 Suppose FX(x) = 1 - 4/x2 for x>2 FX(x) = 0 otherwise a) Find fX(x) We can take the derivative of FX(x) which = 0 - (4)(-2)x-3 = 8/x3 for x>2...
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16.1 Cont Rand Var - cumulative distribution function •...

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