GATE CS 2011 Q33 — Probability: Random Variables and Distributions

We are given a sequence $Y$ derived from $X$ by the linear transformation $y_i = a x_i + b$, where $a, b > 0$. Let us analyze each option to find the incorrect statement:

• Option A (Mode): Since $a > 0$, the transformation $y_i = a x_i + b$ is a one-to-one mapping. The frequency of each element is preserved, meaning the element with the highest frequency (mode) in $X$ will map to the element with the highest frequency in $Y$ at the exact same index position. Thus, this statement is correct.
• Option B (Median): Since $a > 0$, the transformation is strictly increasing (order-preserving). If we sort $X$, the sorted order of indices remains identical for $Y$. Therefore, the median element will be at the same index position in both sequences. Thus, this statement is correct.
• Option C (Mean): The mean of $Y$ is given by: $$\mu_y = \frac{1}{n} \sum_{i=1}^{n} y_i = \frac{1}{n} \sum_{i=1}^{n} (a x_i + b) = a \left(\frac{1}{n} \sum_{i=1}^{n} x_i\right) + b = a \mu_x + b$$ Thus, this statement is correct.
• Option D (Standard Deviation): Standard deviation measures dispersion and is unaffected by shifting all data points by a constant $b$ (translation invariance). Mathematically: $$\sigma_y = \sqrt{\frac{1}{n} \sum_{i=1}^{n} (y_i - \mu_y)^2} = \sqrt{\frac{1}{n} \sum_{i=1}^{n} (a x_i + b - (a \mu_x + b))^2}$$ $$\sigma_y = \sqrt{a^2 \cdot \frac{1}{n} \sum_{i=1}^{n} (x_i - \mu_x)^2} = |a| \sigma_x$$ Since $a > 0$, we have $\sigma_y = a \sigma_x$. The shift $b$ does not affect the standard deviation. Therefore, the statement $\sigma_y = a \sigma_x + b$ is incorrect.