Transforming Random Variables: The PDF of Y-2ln(X) when X~Uniform(0,1)

In probability theory and statistics, the transformation of random variables is a fundamental concept. It involves changing the form of a random variable while preserving its statistical properties. This is particularly important when we want to derive new distributions or simplify complex problems.

Understanding the Problem

The problem we are addressing is: if X ~ Uniform(0,1), what is the probability density function (PDF) of Y -2ln(X)? This question is crucial for understanding how transformations affect the distribution of random variables.

To solve this, we need to apply the transformation technique for random variables. This involves expressing one random variable in terms of the other and using the Jacobian to adjust the density function accordingly.

Step-by-Step Solution

1. Express X in Terms of Y

The first step in the transformation process is to express X as a function of Y. Given Y -2ln(X), we can solve for X:

Y -2ln(X)

ln(X) -Y/2

X exp(-Y/2)

2. Calculate the Jacobian

The Jacobian is the absolute value of the derivative of X with respect to Y. It is a measure of the change in volume as we transform from one coordinate system to another.

J |dx/dy| |d(exp(-Y/2))/dY|

J exp(-Y/2) * (-1/2)

J -1/2 * exp(-Y/2)

Since the Jacobian is always positive, we take the absolute value:

J 1/2 * exp(-Y/2)

3. Combine the PDFs

The PDF of X, since we are given that X ~ Uniform(0,1), is simply 1 for 0 ≤ X ≤ 1. Now, we use the transformation formula:

PDFY(y) PDFX(x) * |J|

Substituting X exp(-Y/2) and |J| 1/2 * exp(-Y/2), we get:

PDFY(y) 1 * (1/2 * exp(-Y/2))

PDFY(y) 1/2 * exp(-Y/2)

Conclusion

Thus, the probability density function of Y -2ln(X) when X follows a Uniform(0,1) distribution is:

PDFY(y) 1/2 * exp(-Y/2)

This function describes the new distribution of Y. Understanding how transformations affect the distribution of random variables is crucial in many areas of statistics, including modeling, simulation, and data analysis.

Further Reading

If you are interested in learning more about random variable transformations and their applications, you may want to refer to the following resources:

Statlect - Transformations of Random Variables MathWorld - Random Variable Transformation arXiv: Random Variable Transformations