Worked solution
Scale one observation or combine independent variables
Independent random variables satisfy and . Find (i) , (ii) , (iii) , (iv) .
CJC · Lecture · Example 7
(i) Method 1: transform the distribution
. Thus to three significant figures. The second normal parameter is variance; use standard deviation for a normalcdf calculation.
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(i) Method 2: transform the event
Alternatively use the original distribution: to three significant figures. Both source methods are retained.
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(ii) Add independent variables
. Consequently to three significant figures.
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(iii) Subtract means but add independent variances
. Rewrite to three significant figures. The variance of a difference is positive and adds here because X and Y are independent.
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(iv) Square the coefficient in the variance
. Hence to three significant figures.
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Common mistakes
Do not subtract variances when subtracting independent variables, or treat 3X as three independent observations.