GATE | GATE CS 2011 | Question 33

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Consider a finite sequence of random values X = { x₁, x₂,…, x_n}. Let μ_x be the mean and σ_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 μ_y be the mean and σ_y be the standard deviation of this sequence. Which one of the following statements is INCORRECT?
(A) Index position of mode of X in X is the same as the index position of mode of Y in Y.
(B) Index position of median of X in X is the same as the index position of median of Y in Y.
(C) μ_y = aμ_x+b
(D) σ_y = aσ_x+b

Answer: (D)

Explanation: Adding a constant like b shift the distribution while multiplying to a constant like a stretch the distribution along median

Mode is the most frequent data of the distribution, so the index position of the mode will not change. From the above graph it is clear that index position of the median will also not change. Now for the mean
$Y_{i} = a X_{i} + b \newline \newline \sum Y_{i} = \sum(aX-{i} + b) \newline \sum Y_{i} = a(\sum X_{i}) + nb \newline (\sum Y_{i})/n = a(\sum X_{i})/n + b \newline \mu _{y} = a\mu_{x} + b$

And for the standard deviation
$\newline \newline \sigma _{y} = \sqrt{\frac{1}{n} \sum (\mu_{y} - Y_{i})^{2}} \newline \newline \sigma _{y} = \sqrt{\frac{1}{n} \sum (a\mu_{x} + b - Y_{i})^{2}} \newline \sigma _{y} = \sqrt{\frac{1}{n} \sum (a\mu_{x} + b - aX_{i} - b)^{2}} \newline \newline \sigma _{y} = \sqrt{\frac{1}{n} \sum (a\mu_{x} - aX_{i} )^{2}} \newline \newline \sigma _{y} = a\sqrt{\frac{1}{n} \sum (\mu_{x} - X_{i} )^{2}} \newline \newline \sigma _{y} = a\sigma_{x}$

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Last Updated : 01 Feb, 2019

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