Answer-first summary for fast verification
Answer: The location shift of $ A $ has no effect on the variance of $ Y $.
## Explanation Let's analyze each option based on the properties of linear transformations: **Given:** Y = A + BX, where B < 0 (decreasing linear transformation) **A. The mean of Y is identical to the mean of X.** - **False:** E[Y] = A + B·E[X], so the mean changes unless A = 0 and B = 1 **B. The location shift of A has no effect on the variance of Y.** - **True:** V[Y] = B²·V[X], so variance depends only on B (the scaling factor), not on A (the location shift) **C. The skewness of Y is identical to the skewness of X.** - **False:** For B < 0, skewness has the same magnitude but opposite sign **D. The kurtosis is affected by decreasing linear transformations.** - **False:** Kurtosis is unaffected by linear transformations (both location and scale changes) Therefore, only statement B is correct.
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Assume that a random variable follows a normal distribution, let , where and are both constant values and is negative, which of the following statements is correct?
A
The mean of is identical to the mean of .
B
The location shift of has no effect on the variance of .
C
The skewness of is identical to the skewness of .
D
The kurtosis is affected by decreasing linear transformations.