Problem 178 - An Introduction to Mathematical Statistics and Its Applications

Question

Prove that if \(X\) and \(Y\) are two independent random variables, then \(U=g(X)\) and \(V=h(Y)\) are also independent.

Step-by-Step Solution

Answer

If \(X\) and \(Y\) are independent, then any functions of \(X\) and \(Y\), denoted as \(U = g(X)\) and \(V = h(Y)\), are also independent. This is confirmed by substituting \(U = g(X)\) and \(V = h(Y)\) into the independence formula to find that the joint probability mass (or density) function of \(U\) and \(V\) equals the product of their individual mass (or density) functions for all values, which is the definition of independence.

Step 1: Define the Random Variables and Functions

Let \(X\) and \(Y\) be two independent random variables and \(g\) and \(h\) be any functions. Define \(U = g(X)\) and \(V = h(Y)\). The task is to prove that \(U\) and \(V\) are also independent.

Step 2: Understand the Definition of Independence

Remember that, by definition, two random variables \(A\) and \(B\) are independent if the joint probability mass function is the product of their individual mass functions, that is, \(P(A, B) = P(A)P(B)\) for all values of \(A\) and \(B\). If the random variables are continuous, the same holds for their density functions.

Step 3: Apply the Definition of Independence to the Functions

Apply the definition of independence to \(U\) and \(V\). By definition, \(U\) and \(V\) are independent if the joint probability mass (or density) function of \(U\) and \(V\) equals the product of their individual mass (or density) functions for all values.

Step 4: Substitute \(U\) and \(V\) By Their Definitions

Substitute \(U = g(X)\) and \(V = h(Y)\) into the formula. Because \(X\) and \(Y\) are independent, their joint mass (or density) function equals the product of their individual mass (or density) functions. This confirms that the joint mass (or density) function of \(U\) and \(V\) equals the product of their individual mass (or density) functions for all values, proving that \(U\) and \(V\) are independent.

Key Concepts

Independent Random Variables

Understanding the concept of independent random variables is fundamental in probability theory. Two random variables, say \(X\) and \(Y\), are considered independent if the probability of their joint occurrence is equivalent to the product of their individual probabilities. Mathematically, for all possible values of \(x\) and \(y\), this is expressed as:\[P(X = x, Y = y) = P(X = x) \cdot P(Y = y)\]This independence implies that the behavior or outcome of one random variable does not influence or alter the outcome of the other. In practice, this can be visualized as events happening simultaneously yet distinctly.

• Example: Rolling two dice: The result of the first die does not affect the result of the second.
• Significance: Independent variables simplify calculations of probabilities and expectations.

In simpler terms, the occurrence of one event does not provide any information about the likelihood of the occurrence of the other. This concept is crucial when dealing with transformations of such variables.

Transformation of Variables

Transformation of variables is a technique where we apply functions to random variables to create new variables. If \(X\) and \(Y\) are independent random variables, and we define \(U = g(X)\) and \(V = h(Y)\) where \(g\) and \(h\) are functions, the goal is to explore if \(U\) and \(V\) are also independent.The transformation leverages functions to map one set of values to another. Importantly, in probability, this concept is beneficial because it allows for generalizations where variables can be manipulated yet preserve properties like independence.

• Procedure: Choose functions \(g\) and \(h\) that map random variables \(X\) and \(Y\) to \(U\) and \(V\) respectively.
• Key Idea: Despite transformations, if \(X\) and \(Y\) are initially independent, \(U\) and \(V\) will typically remain independent.

One clever use of this is when working with distributions, such as applying logarithmic or exponential functions to normal distributions to model different phenomena while maintaining variable independence.

Joint Probability Density Function

A joint probability density function (pdf) describes the likelihood of two continuous random variables occurring in tandem. For two random variables \(X\) and \(Y\), the joint pdf is denoted as \(f_{X,Y}(x, y)\).

• Continuous Variables: Used for continuous data as opposed to discrete data, which uses probability mass functions.
• Mathematical Expression: The joint pdf for independent variables is the product of their individual densities: \[f_{X,Y}(x, y) = f_X(x) \cdot f_Y(y)\]

This concept serves as a cornerstone for analyzing the behavior and properties of multiple variables in the same probabilistic framework. It allows us to understand, compute, and predict the joint occurrence of events particularly when independent transformations of variables are considered.