Independent solution

How to solve this Linear Combinations of Independent Random Variables question

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)}}