Let (X, Y) be a pair of random variables with joint density...

Let $(X, Y)$ be a pair of random variables with joint density function $$ f_{X, Y}(x, y)= \begin{cases}\frac{2|x|+y}{\sqrt{2 \pi}} \exp \left\{-\frac{(2|x|+y)^2}{2}\right\} & \text { if } y \geq-|x|, \\ 0 & \text { if } y<-|x| .\end{cases} $$ Show that $X$ and $Y$ are standard normal random variables and that they are uncorrelated but not independent.

Let $(X, Y)$ be a pair of random variables with joint density function
$$
f_{X, Y}(x, y)= \begin{cases}\frac{2|x|+y}{\sqrt{2 \pi}} \exp \left\{-\frac{(2|x|+y)^2}{2}\right\} & \text { if } y \geq-|x|, \\ 0 & \text { if } y<-|x| .\end{cases}
$$

Show that $X$ and $Y$ are standard normal random variables and that they are uncorrelated but not independent.

Key Concepts

Joint Density Function

This concept refers to the function that provides the probability density for a pair (or more) of random variables simultaneously. In the context of the question, the joint density function defines how the two random variables (X, Y) interact and what regions in their two-dimensional domain are assigned positive probability, which is essential for analyzing their individual (marginal) behaviors and any potential dependence between them.

Marginal Distributions

Marginal distributions are obtained by integrating the joint density function over the other variable. This process isolates the behavior of one random variable irrespective of the other. In the problem, computing the marginal distributions for X and Y separately shows that each has a standard normal distribution, meaning they follow a normal distribution with mean zero and variance one.

Standard Normal Distribution

A standard normal distribution is a normal probability distribution with a mean of zero and a variance of one. It is highly significant in probability theory and statistics because of its well-known properties, including its symmetric bell shape and role in the central limit theorem. Demonstrating that X and Y are standard normal random variables involves showing that their marginal densities match that of a standard normal.

Uncorrelated Random Variables

Two random variables are said to be uncorrelated if their covariance is zero. Covariance being zero indicates that there is no linear relationship between the variables. In this problem, while X and Y are uncorrelated, meaning their linear association is absent, this does not imply that they are statistically independent.

Statistical Independence

Independence between random variables means that the occurrence of one does not affect the probability distribution of the other; mathematically, their joint density factors into the product of their marginal densities. Although X and Y have marginal distributions that are standard normal and are uncorrelated (zero covariance), the joint density function does not factorize accordingly, indicating that there exists a form of dependence between them beyond simple linear correlation.

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