Financial Risk Manager Part 1

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Explanation:

When a random variable undergoes a linear transformation $Y = aX + b$ :

Location measures (mean, median) are both scaled by $a$ and shifted by $b$ .
Spread or dispersion measures like Variance are scaled by $a^2$ ( $Var(Y) = a^2Var(X)$ ). The Interquartile Range (IQR) and Standard Deviation are mathematically scaled by $|a|$ , though the option groups variance and standard deviation under $a^2$ (a common slight imprecision in questions, but conceptually separating shift vs. scale).
Shape parameters (skewness and kurtosis) are standardized moments and are completely unaffected by shifting and scaling (provided $a > 0$ ). Option B is the only logically viable answer ruling out the false shifts in spread/shape present in A, C, and D.

Explanation:

When a random variable undergoes a linear transformation $Y = aX + b$ :

Location measures (mean, median) are both scaled by $a$ and shifted by $b$ .
Spread or dispersion measures like Variance are scaled by $a^2$ ( $Var(Y) = a^2Var(X)$ ). The Interquartile Range (IQR) and Standard Deviation are mathematically scaled by $|a|$ , though the option groups variance and standard deviation under $a^2$ (a common slight imprecision in questions, but conceptually separating shift vs. scale).
Shape parameters (skewness and kurtosis) are standardized moments and are completely unaffected by shifting and scaling (provided $a > 0$ ). Option B is the only logically viable answer ruling out the false shifts in spread/shape present in A, C, and D.

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Q.73 Consider a random variable X with a known probability distribution. Suppose we define a new random variable Y as a linear transformation of X, such that Y = aX + b, where a and b are constants. Which of the following statements correctly describes the effect of this linear transformation on the statistical properties of X?

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Last updated: May 21, 2026 at 13:04

The mean, median, and interquartile range of Y are all scaled by a and shifted by b, while the variance, standard deviation, skewness, and kurtosis are unaffected.

The mean and median of Y are scaled by a and shifted by b; the variance and standard deviation are scaled by a²; the skewness and kurtosis remain unchanged; and the interquartile range is scaled by a.

The mean, variance, and standard deviation of Y are scaled by a and shifted by b, while the skewness, kurtosis, median, and interquartile range remain unaffected.

The mean, variance, standard deviation, skewness, kurtosis, median, and interquartile range of Y are all scaled by a and shifted by b.