If $Y = 5 - X$, what is the relationship between $\sigma_Y$ and $\sigma_X$?

$\sigma_Y = \sigma_X$. The multiplier $b$ is $-1$, and $|-1| = 1$, so the standard deviation remains the same.

How does adding a constant $a$ to every value in a random variable affect its standard deviation?

It has no effect. Adding a constant shifts the entire distribution but does not change the distance between values, so the spread remains identical.

When transforming $X$ into $Y = a + bX$, why is the absolute value of $b$ used for the new standard deviation?

Standard deviation measures distance and must be non-negative. If $b$ is negative, the absolute value ensures the resulting spread is a positive value.

What is the difference between the transformation of the mean and the transformation of the variance?

The mean is affected by both addition ($a$) and multiplication ($b$), while the variance is only affected by the square of the multiplier ($b^2$).

If a distribution is skewed right and you multiply every value by $-2$, what happens to the shape?

The distribution is flipped horizontally. Because of the negative multiplier, the right skew becomes a left skew.

What is the most common mistake when calculating the variance of $Y = 10 - 4X$?

Forgetting to square the multiplier $(-4)$ or accidentally including the constant $10$ in the calculation. The correct multiplier for variance is $(-4)^2 = 16$.

Does a linear transformation $Y = a + bX$ change the shape of a Normal distribution?

No. A linear transformation of a Normal random variable results in another Normal random variable, though its center and spread will change.

Define the formula for the mean of a transformed random variable $Y = a + bX$.

$\mu_Y = a + b\mu_X$. This shows that the mean follows the exact same linear path as the individual data points.

Define the formula for the standard deviation of $Y = a + bX$.

$\sigma_Y = |b|\sigma_X$. The constant $a$ is ignored because shifts do not affect variability.

If you divide every value in a dataset by 2, what happens to the variance?

The variance is divided by 4. Since dividing by 2 is the same as multiplying by $b = 0.5$, the variance is multiplied by $b^2 = 0.25$.

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Statistics

Unit 4: Probability, Random Variables & Probability Distributions

Linear Transformations of Random Variables

Summary

Linear transformations involve modifying a random variable by adding a constant, multiplying by a constant, or both. These operations systematically alter the center, spread, and sometimes the shape of the probability distribution, following specific mathematical rules that distinguish between measures of position and measures of variability.

1. Definition & Core Concepts

A linear transformation of a random variable $X$ is a new random variable $Y$ defined by the linear equation $Y = a + bX$ , where $a$ and $b$ are real numbers.
The constant $a$ represents a shift or translation (addition/subtraction), while the constant $b$ represents a scaling factor (multiplication/division).
These transformations are commonly used in statistics for unit conversions (e.g., Celsius to Fahrenheit) or to rescale data for better interpretability.

Diagram showing how addition shifts a distribution horizontally while multiplication changes its width and height.

2. Impact on Measures of Center

3. Impact on Measures of Spread

4. Effects on Distribution Shape

5. Key Distinctions

6. Exam Strategy & Tips