Determining the Distributions for Which the Product XY is Standard Normal

When considering the product of two random variables X and Y and wanting this product to be a standard normal distribution, we encounter a complex problem that hinges on the distributions of X and Y. Let's delve into the mathematical intricacies and possible solutions.

The Problem at Hand

The primary question is, which distributions should X and Y belong to in order for their product XY to follow a standard normal distribution? This problem assumes independence between X and Y, denoted as X and Y being i.i.d. (independent and identically distributed).

Investigating the Relationship Between X, Y, and U ln(XY) ln(X) ln(Y)

A helpful substitution in approaching this problem is to consider U ln(XY) ln(X) ln(Y). This transformation simplifies the relationship between X, Y, and U. Given that XY e^U, we can derive a distribution for U and analyze its properties.

From the derived distribution of U, fu sqrt{frac2{pi}}e^{-frac12e^{2u}}e^{-u}, it is evident that this distribution is skewed to the left. This skewness points towards a close relationship with the Gumbel distribution, a well-known type of heavy-tailed distribution used in extreme value theory.

Implications of the Derived Distribution for U

The skewness of the distribution of U implies that the logarithm of the product XY is not easily invertible to a standard normal distribution. This skewness indicates that the direct transformation from the distribution of U to a standard normal distribution is non-trivial and not likely to yield a simple closed form solution.

Alternative Approaches and Considerations

One possible approach to solving this problem is to consider the characteristic function of X and Y. Given the independence and identical distribution assumption, the characteristic function can be obtained, and the inversion theorem can be applied to find the distribution of XY. However, this method may not yield a simple closed-form solution, especially given the complexity of the product of random variables.

Specific Solutions for X and a Constant Y

A simpler solution is to set X as a normal distribution with mean zero and Y as a constant. This specific setup ensures that XY indeed follows a standard normal distribution. This example demonstrates that while finding a general closed-form solution for arbitrary distributions of X and Y is challenging, specific cases can provide simple and tractable solutions.

Conclusion and Further Reading

In conclusion, while there isn't a general closed-form solution for arbitrary distributions of X and Y that guarantees XY will be standard normal, special cases can provide simple solutions. The skewness of the distribution of ln(XY) highlights the complexity of the problem, and methods such as characteristic functions offer further insights but may not always yield closed-form solutions.

We hope this exploration of the problem provides valuable insights into the relationships between distributions and products of random variables. For further reading, consider exploring the Gumbel distribution and its properties, as well as the application of characteristic functions in probability theory.