# TP%2310_f08 - the form Y=a bX By definition the expected...

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CE 3102 Fall 2008 Thought Problem 10 Temperature Conversion You’ve made a set of measurements of a lake’s temperature, and found that they appear to vary randomly from day to day, with a mean of 12 o Celsius, and a variance of 2. You plan to present these findings at a public meeting in the USA, and want to convert these to Fahrenheit. Unfortunately, you don’t have ready access to the original measurements. What can you do?

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CE 3102 Fall 2008 Thought Problem 10 Instructor’s Partial Solution The place to start is with the Celsius to Fahrenheit conversion: F = (9/5)C +32, One option would be to convert each of your individual data items to Fahrenheit, and then compute their mean and variance, but you don’t have these available. A more elegant approach would be to note that this is an example of a linear transformation, of
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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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TP%2310_f08 - the form Y=a bX By definition the expected...

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