ch6sometechniquesofcalculation-2.studentview

# Often we are interested in a linear transformation of

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Unformatted text preview: N = 15 µ = 156 σ = 345 Data Find the mean and s.d. of the 25 numbers. 2 2 2 2 N1 = 10 µ1 = 141 σ1 =15 B. Linear Transformation B. Consider the data set x1 , x2 ,......., xN Consider 2 µ X and σ X Suppose their mean and variance are respectively. Often we are interested in a linear transformation of them: yi = a + bxi ............(1) them: Thus, we have new data: y1 , y2 ,......., y N For instance x1 , x2 ,......., xN may be temperatures in For centigrade. Now we like to express them in Fahrenheit: Fahrenheit: y = 32 + 9 x i 5 i We like to ask: What are the mean µ and variance these new data y = a + bx ? these Y 2 σY of i i The answers are: µ =a +bµ ............( 2) Y X 2 σ2 =b 2σX ................(3) Y σ = b | × X ..............( 4) | σ Y Proof: From (1), From yi = a + bxi µY = a + bµ X yi − µY = a + bxi − a − bµ X ∑y i =1 N i =1 N i = Na + b∑ xi i =1 N ∑(y −...
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