Independent solution

How to solve this Linear Combinations of Independent Random Variables question

Answer in brief

Each observation has variance 9. Independence makes the variance of the linear combination equal to 3^2(9)+9+9+9=108, so choice E is correct.

Setup

Setup

Convert the common standard deviation into variance.

Var⁡(Xi)=32=9\operatorname{Var}(X_i)=3^2=9

Model

Model

For independent observations, square every coefficient and add the resulting variance contributions.

Var⁡(3X1−X2−X3−X4)\operatorname{Var}(3X_1-X_2-X_3-X_4)
=32Var⁡(X1)+∑i=24(−1)2Var⁡(Xi)=3^2\operatorname{Var}(X_1)+\sum_{i=2}^{4}(-1)^2\operatorname{Var}(X_i)

Compute

Compute

Substitute the common variance.

Var⁡(3X1−X2−X3−X4)=9(9)+9+9+9=108\operatorname{Var}(3X_1-X_2-X_3-X_4)=9(9)+9+9+9=108

Answer

Answer

The variance is 108.

108(E)\boxed{108\quad\text{(E)}}