Discussion
Let X and Y be random variable with the following distributions respectively :
| X | 1 | 2 | 3 | 4 | 5 |
| P | 0.1 | 0.2 | 0.4 | 0.2 | 0.1 |
| Y | 10 | 20 | 35 | 50 | 60 |
| P | 0.1 | 0.2 | 0.4 | 0.2 | 0.1 |
Answer: C
The skewness is the same, but the coefficients of variation are not.
Both distributions have the probabilities 0.1, 0.2, 0.4, 0.2, 0.1 placed symmetrically about their middle value: X about 3, and Y about 35 (35 is 15 above 20 and 15 below 50, 25 above 10 and 25 below 60). A symmetric distribution has zero skewness, so the skewness is equal.
\[ \begin{aligned} \sigma_X^2 &= 2(0.1 \cdot 4) + 2(0.2 \cdot 1) = 1.2 \\ CV_X &= \sqrt{1.2}/3 \approx 0.365 \\ \sigma_Y^2 &= 2(0.1 \cdot 625) + 2(0.2 \cdot 225) = 215 \\ CV_Y &= \sqrt{215}/35 \approx 0.419 \end{aligned} \]
The coefficients of variation differ because Y is not a rescaling of X (its values are not evenly spaced). That rules out every option saying the coefficients are the same, and every option saying the skewness differs.
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