Answer: The location shift of $ A $ has no effect on the variance of $ Y $.

## Explanation Let's analyze each option: **A. The mean of $ Y $ is identical to the mean of $ X $.** - **Incorrect**: From the linear transformation formula: $ E[Y] = A + BE[X] $ - Since $ A \neq 0 $ and $ B \neq 1 $, the mean of Y is different from the mean of X **B. The location shift of $ A $ has no effect on the variance of $ Y $.** - **Correct**: Variance formula: $ V[Y] = B^2V[X] $ - The variance depends only on $ B^2 $, not on the location shift parameter $ A $ - This is a fundamental property of variance - location shifts don't affect dispersion **C. The skewness of $ Y $ is identical to the skewness of $ X $.** - **Incorrect**: When $ B < 0 $ (decreasing transformation), skewness has the same magnitude but opposite sign - Since X follows a normal distribution, its skewness is 0, so Y's skewness would also be 0 - However, the statement is technically incorrect because it doesn't account for the sign change **D. The kurtosis is affected by decreasing linear transformations.** - **Incorrect**: Kurtosis is unaffected by linear transformations (both increasing and decreasing) - For normal distributions, kurtosis remains 3 regardless of linear transformations **Key Insight**: For linear transformations $ Y = A + BX $: - Mean: $ E[Y] = A + BE[X] $ - Variance: $ V[Y] = B^2V[X] $ (location shift A has no effect) - Skewness: Same magnitude, sign changes if $ B < 0 $ - Kurtosis: Unaffected by linear transformations

Author: LeetQuiz .