17LinearCombos

17LinearCombos - Statistical Techniques I EXST7005 Linear...

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Statistical Techniques I EXST7005 Linear Combinations
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Linear combinations This is a function of random variables of the form Σ aiYi where ai is a constant and Yi is a variable. Generic Example: We want to create a score we can use to evaluate students applying to LSU as freshmen. Score=a(VerbalSAT)+b(MathSAT)+c(GPA) where a, b and c are the constants and VerbalSAT, MathSAT and GPA are the variables (they vary among students).
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Linear combinations (continued) Score=a(VerbalSAT)+b(MathSAT)+c(GPA) we need to choose values of a, b and c if a=1/3 and b=1/3 and c=1/3 then we have an average (a+b+c)/3. If a=1 and b=1 and c=1 we have a simple sum (a+b+c). But VerbalSAT and MathSAT are values in the hundreds and GPA is around 2 or 3. So we might choose a = b = 1/100 and c = 1. Any of these is a linear combination.
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Mean and Variance So what is the mean value of our linear combination, and can we put a variance on i (to get a confidence interval)? Linear combination: Σ aiYi Expected value: Σ ai μ Yi Estimate of mean: Σ ai f8e5 Yi
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Mean and Variance (continued) Variance of the linear combination. The variance of a linear combination is the sum of the variances of the individual variances (wi squared coefficients) plus twice the covariance of the variables (with both coefficients). Estimate of the variance: Σ a2iSi2 + 2aiaj(Covariances)
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Mean and Variance (continued) For example, with the Score we calculated previously, the variance might be VAR(Score) = a2VAR(Verbal) + b2VAR(Math) + c2VAR(GPA) + (2abCOV(Verbal,Math) + 2ac(Verbal,GPA) + 2bcCOV(Math,GPA)
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Mean and Variance (continued) HOWEVER, if the variables are independent the covariance can assumed to be zero. The linear combination reduces to the sum o
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17LinearCombos - Statistical Techniques I EXST7005 Linear...

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