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Unformatted text preview: the form Y=a+bX By definition, the expected value for Y is âˆ« + = + = dx x f bx a bX a E Y E ) ( ) ( ] [ ] [ where f(x) is the pdf for X. Using the linearity properties of integration then gives us âˆ« âˆ« âˆ« âˆ« âˆ« + = + = + = + ] [ ) ( ) ( ) ( ) ( ) ( ) ( ) ( X bE a dx x xf b dx x f a dx x f bx dx x af dx x f bx a Similarly, the variance for Y, by definition, would be âˆ« +-+ = +-+ = dx x f b a bx a b a bX a E Y V ) ( )) ( ) (( )] ( ) [( ] [ 2 2 Î¼ where the symbol Î¼ is used for E[X]. Some canceling and factoring then gives us âˆ« âˆ« =-=-= ] [ ) ( ) ( ) ( )) ( ( ] [ 2 2 2 2 X V b dx x f x b dx x f x b Y V So, our mean temperature, in Fahrenheit, would be (9/5)(12)+32 = 53.6 o F and the variance, in Fahrenheit, would be (9/5) 2 (2) = 6.48...
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- Spring '11
- Variance, Probability theory, Fahrenheit, Celsius, Cauchy distribution, Thought Problem