Variance of Summations Let X1,X2,.,Xn be random variables with finite variance,then we have var(∑iXi)=∑ivar(Xi)+∑i=jCOV(Xi,Xi) 6
Variance of Summations • Let 𝑋1, 𝑋2, ⋯ , 𝑋𝑛 be random variables with finite variance, then we have var σ𝑖 𝑋𝑖 = σ𝑖 var(𝑋𝑖) + σ𝑖≠𝑗 cov(𝑋𝑖 , 𝑋𝑗) 6
Example 2 If the variance of verbal GRE were 64,the variance of quantitative SAT were 81 and the correlation between these two tests were 0.50,then what is the variance of total SAT(verbal quantitative)? 7
Example 2 • If the variance of verbal GRE were 64, the variance of quantitative SAT were 81 and the correlation between these two tests were 0.50, then what is the variance of total SAT (verbal + quantitative)? 7
Example 2 Denote X1,X2 as the score for verbal and quantitative respectively, then var(X1+X2)=var(X1)+var(X2)+2cov(X1,X2) =64+81+2×0.5×V64×V81 =217 8
Example 2 • Denote 𝑋1, 𝑋2 as the score for verbal and quantitative respectively, then var 𝑋1 + 𝑋2 = var 𝑋1 + var 𝑋2 + 2cov 𝑋1, 𝑋2 = 64 + 81 + 2 × 0.5 × 64 × 81 = 217 8
Conditional Expectation Revisit The conditional expectation EY of a random variable X given another random variable y,is a new random variable determined by Y. It's distribution is determined by the distribution of Y
Conditional Expectation Revisit • The conditional expectation E[𝑋|𝑌] of a random variable 𝑋 given another random variable 𝑌, is a new random variable determined by 𝑌. • It’s distribution is determined by the distribution of 𝑌