GATE CS 2011 Q.33 (CS Q.33): Probability and Statistics

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?

Index position of mode of $X$ in $X$ is the same as the index position of mode of $Y$ in $Y$ .

Index position of median of $X$ in $X$ is the same as the index position of median of $Y$ in $Y$ .

$\mu_y = a \mu_x + b$

$\sigma_y = a \sigma_x + b$

Official answer and solution

Official answer: D
From: Official GATE 2011 answer key

Solution

Recall the transformation properties of summary statistics under a linear shift and scale $Y = a X + b$ .

Because $a > 0$ , the linear transformation $y_i = a x_i + b$ is strictly monotonically increasing. Hence, order-based statistics preserve ranks: the index positions of the mode (most frequent value) and the median (middle sorted value) remain unchanged in the sequence (Statements A and B are true).

By linearity of expectation, the mean transforms as $\mu_y = E[a X + b] = a E[X] + b = a \mu_x + b$ (Statement C is true).

The variance scales only by $a^2$ : $\text{Var}(Y) = \text{Var}(a X + b) = a^2 \text{Var}(X)$ . Taking the square root gives standard deviation $\sigma_y = a \sigma_x$ .

Standard deviation is invariant to constant additive shifts $b$ . Statement D claims $\sigma_y = a \sigma_x + b$ , which is mathematically INCORRECT.

Final answer

The answer is the one in the official GATE answer key. The solution is this site's own explanation.

Source

Paper: GATE 2011 Computer Science and Information Technology (CS)
Question: 33 of 65, printed as CS Q.33
Section: Computer Science and Information Technology

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