Introduction

In the world of statistics and probability, understanding the impact of transformations on random variables is crucial for making informed decisions. In this chapter, we will delve into the concept of linear transformations and explore how they affect various statistical properties of random variables. Linear transformations play a significant role in real-world applications, enabling us to manipulate data and draw meaningful insights. We will discuss how changes in scale and location through linear transformations influence the mean, variance, standard deviation, skewness, kurtosis, median, and interquartile range of random variables.

Effect of Linear Transformation on Statistical Properties

Linear transformations involve multiplying a random variable by a constant and then adding another constant. If $X$ is a random variable, a linear transformation can be represented as $Y = aX + b$, where ‘$a$’ is the scaling factor and ‘$b$’ is the translation factor. Let’s explore the effects of this transformation on various statistical properties:

• Mean: The mean of the transformed variable $Y$ is affected by both ‘$a$’ and ‘$b$’. The new mean of $Y$ is given by $E(Y)=aE(X)+b$.

• Variance and Standard Deviation: The variance and standard deviation of $Y$ are affected by the scaling factor ‘$a$’. The new variance of $Y$ is $Var(Y)=a^2Var(X)$, and the standard deviation becomes $SD(Y)=\mid a \mid SD(X)$.

• Skewness and Kurtosis: The skewness and kurtosis of $Y$ are not affected by linear transformations. These properties remain unchanged.

• Median and Interquartile Range: Linear transformations do not affect the relative ordering of data. Therefore, the median and interquartile range remain the same after the transformation.

Example: Let’s consider a dataset of exam scores $(X)$ with a mean of 75 and a standard deviation of 10. If we apply a linear transformation $Y=2X+5$, the new mean becomes $155(2 \times 75+5)$ and the new standard deviation becomes 20 ($\mid 2 \mid \times 10$).