Suppose that the range of the continuous variables X and Y...

Suppose that the range of the continuous variables $X$ and $Y$ is $0<x<a$ and $0 < y < b$. Also suppose that the joint probability density function $f_{X Y}(x, y)=g(x) h(y),$ where $g(x)$ is a function only of $x$ and $h(y),$ is a function only of $y .$ Show that $X$ and $Y$ are independent.

Key Concepts

Independence of Random Variables

Two random variables are independent if and only if the joint probability density function can be decomposed into the product of their marginal probability density functions. This factorization property indicates that the occurrence of one event gives no information about the occurrence of the other, which is a fundamental concept in probability theory.

Marginal Probability Density Function

The marginal probability density function of a continuous random variable is obtained by integrating the joint probability density function over the range of the other variable. This process provides the standalone distribution for each variable, ignoring any dependence between them.

Probability Density Function (pdf)

A probability density function is a function that describes the likelihood of a continuous random variable taking a particular value. It has the property that the probability of the variable falling within a specific interval can be evaluated by integrating the pdf over that interval, and the total area under the curve equals one.

Joint Probability Density Function

The joint probability density function of two continuous random variables expresses the probability that each variable falls within a certain range simultaneously. This function encapsulates the relationship between the variables and is critical for determining their dependence or independence.

Continuous Random Variables

Continuous random variables are those that can take any value within a given interval, and they are described by probability density functions (pdfs) instead of probability mass functions. Understanding continuous random variables requires familiarity with how their outcomes are modeled over intervals and how integration is used to calculate probabilities.

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