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Unformatted text preview: cumulative distribution function : â€¢ Which is the area under the curve over the range Xx Â± âˆ« âˆž= = â‰¤ x X du u f x F x X P ) ( ) ( ) ( 1/5/10 Mean and Variance â€¢ The following definitions apply: [ ] âˆ« âˆ« âˆ« âˆ« âˆž âˆžâˆž âˆžâˆž âˆžâˆž âˆž== = = = 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) = 2x 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)x3 = 8/x3 for x>2...
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 Spring '08
 Rodriguez
 Statistics, Probability, Probability distribution, Probability theory, probability density function, Cumulative distribution function, Continuous probability distribution

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