If X and Y are independent standard normal random variables,...

If $X$ and $Y$ are independent standard normal random variables, determine the joint density function of
$$
U=X \quad V=\frac{X}{Y}
$$
Then use your result to show that $X / Y$ has a Cauchy distribution.

Key Concepts

Ratio Distribution and Cauchy Distribution

The ratio distribution arises when one random variable is divided by another. In the given problem, the variable V is defined as X/Y, which represents the ratio of two independent standard normals. A classic result in probability theory is that this ratio follows a Cauchy distribution. The Cauchy distribution is known for its heavy tails and the non-existence of finite moments, and recognizing this result is central to understanding the behavior of ratios in the context of normal distributions.

Standard Normal Distribution

A standard normal distribution is a fundamental probability distribution in statistics characterized by a mean of 0 and a variance of 1. In the problem, both X and Y are independent standard normal random variables. Their well-known density functions greatly simplify the analysis when these variables are transformed, as their properties are exploited in computing the joint density of the transformed variables.

Jacobian Determinant

When changing variables in a multivariate probability density function, the Jacobian determinant of the transformation plays a crucial role. It accounts for the scaling of volume elements during the transformation. In the presented problem, the inverse transformation is used to compute the determinant of the matrix of partial derivatives, ensuring that the new density is correctly adjusted from the original joint density of X and Y.

Transformation of Random Variables

This concept refers to the method of finding the probability density function of new random variables that are defined as functions of original random variables. In the problem, the original variables X and Y are transformed into U and V where U = X and V = X/Y. This involves determining the relationship between the old and new variables and then applying the transformation theorem to obtain the joint density of the new variables.

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