The Cauchy-Schwartz inequality implies that the asymptotic variance of is:
A)greater than .
B)less than or equal to .
C) equal to .
D) less than .

Question

The Cauchy-Schwartz inequality implies that the asymptotic variance of ​  is:
A)​greater than . 
B)​less than or equal to . 
C) ​equal to . 
D) ​less than .

Quizplus · Accepted Answer

The Cauchy-Schwartz inequality states that for any two random variables X and Y, the following inequality holds:

|Cov(X,Y)| ≤ √(Var(X)Var(Y))

In this case, X is 11eab06d_250f_af02_beed_87b0fdfcd9e6_TB2133_11 and Y is 1. We know that the variance of Y is 0, since it is a constant. Therefore, we can simplify the inequality to:

|Cov(11eab06d_250f_af02_beed_87b0fdfcd9e6_TB2133_11,1)| ≤ √(Var(11eab06d_250f_af02_beed_87b0fdfcd9e6_TB2133_11) * 0)

The covariance between two independent random variables is 0, so we can further simplify to:

|Corr(11eab06d_250f_af02_beed_87b0fdfcd9e6_TB2133_11,1)| ≤ √Var(11eab06d_250f_af02_beed_87b0fdfcd9e6_TB2133_11)

The correlation between 11eab06d_250f_af02_beed_87b0fdfcd9e6_TB2133_11 and 1 is 1, since 1 is a constant. Therefore, we can simplify further to:

1 ≤ √Var(11eab06d_250f_af02_beed_87b0fdfcd9e6_TB2133_11)

Squaring both sides gives:

1 ≤ Var(11eab06d_250f_af02_beed_87b0fdfcd9e6_TB2133_11)

Therefore, the asymptotic variance of 11eab06d_250f_af02_beed_87b0fdfcd9e6_TB2133_11 is less than or equal to 1, and the best choice is B. The Cauchy-Schwartz inequality implies that the asymptotic variance of 11eab06d_2510_2437_beed_9f56e623ae51_TB2133_11 is less than or equal to 11eab06d_2510_4b48_beed_4f018129c482_TB2133_11 .

The Cauchy-Schwartz Inequality Implies That the Asymptotic Variance of ​

The Cauchy-Schwartz Inequality Implies That the Asymptotic Variance of