Understanding the Expected Value and Variance of a Transformed Random Variable
In this article, we will explore how to calculate the expected value (EY) and variance (VY) of a transformed random variable ( Y 6X - 3 ), given that the original random variable ( X ) has an expected value ( EX 3 ) and a variance ( VX 7 ). We will discuss the properties of expected values and variances and apply these properties to solve the problem step-by-step.
Properties of Expected Values and Variances
We start with two fundamental properties of expected values and variances:
Expected Value Property: For any constants ( a ) and ( b ), the expected value of the random variable ( aX b ) is given by:
Variance Property: For any constant ( a ), the variance of the random variable ( aX b ) is given by:
Mathematically, these properties can be written as:
Expected Value Property:
For any constants ( a ) and ( b ),
[ E[aX b] aE[X] b ]
Variance Property:
For any constant ( a ),
[ Var[aX b] a^2Var[X] ]
These properties ensure that both expected values and variances behave in a linear manner under additive and multiplicative operations with constants.
Calculation of EY and VY
Given that ( EX 3 ) and ( VX 7 ), we need to determine ( EY ) and ( VY ) for ( Y 6X - 3 ).
To find ( EY ), we use the expected value property:
[ EY E[6X - 3] 6EX - 3 ]
Substituting the given value of ( EX ):
[ EY 6 times 3 - 3 18 - 3 21 ]
Next, to find ( VY ), we use the variance property:
[ VY Var[6X - 3] 6^2 VX ]
Substituting the given value of ( VX ):
[ VY 6^2 times 7 36 times 7 252 ]
Thus, we have calculated the expected value and variance for the transformed random variable ( Y ):
Expected Value ( EY 21 )
Variance ( VY 252 )
Deeper Insight: Linear Properties of Expectations and Variances
The properties of expected values and variances have much broader implications. They can be applied to any linear transformation of a random variable. The intuition behind these properties is that the expected value (or mean) of a transformed random variable is simply the transformation applied to the expected value of the original variable, and the variance of a scaled random variable is the square of the scaling factor times the variance of the original variable.
These properties extend to integrals of functions with respect to a probability measure, ensuring that expectations and variances behave consistently across different types of random variables, whether discrete or continuous. Specifically:
For discrete random variables, the expected value is a weighted sum of possible outcomes.
For continuous random variables, the expected value is an integral of the product of the variable and its probability density function.
Summary
In summary, the expected value and variance of the transformed random variable ( Y 6X - 3 ) are calculated as follows:
Expected Value ( EY 21 )
Variance ( VY 252 )
This article provides a detailed explanation of the underlying properties and steps involved in these calculations. Whether you are working with discrete or continuous random variables, understanding these linear properties is fundamental to working with transformed random variables in probability and statistics.
Recommended Further Reading
For a deeper understanding of these concepts, consider exploring further topics in probability theory and statistics, including:
Random Variables and Distributions
Transformations of Random Variables
Probability Theory
Statistical Inference
By mastering these foundational concepts, you can build a strong foundation for advanced studies in data science, machine learning, and other quantitative fields that rely on probabilistic models and random variables.