Consider a finite sequence of random values $X=[x_1,x_2,\dots x_n]$. Let $\mu_x$ be the mean and $\sigma_x$ be the standard deviation of $X$. Let another finite sequence $Y$ of equal length be derived from this as $y_i=a*x_i+b$, where $a$ and $b$ are positive constants. Let $\mu_y$ be the mean and $\sigma_y$ be the standard deviation of this sequence. Which one of the following statements is INCORRECT?
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ndex position of mode of $X$ in $X$ is the same as the index position of mode of $Y$ in $Y$
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ndex position of median of $X$ in $X$ is the same as the index position of median of $Y$ in $Y$
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$\mu_y=a \mu_x + b$
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$\sigma_y=a \sigma_x + b$
D
Correct answer
Explanation
For linear transformation Y = aX + b: mean scales as μ_y = aμ_x + b (shifts and stretches). Standard deviation scales as σ_y = aσ_x (only stretches, not shifts). Option D incorrectly adds b to standard deviation - this is wrong because b is a constant shift and doesn't affect spread. Options A and B about mode/median index positions are correct (order statistics preserved under linear transforms). Option C about mean is correct.