Yes, your statement is correct.

Given that $X$ follows a $\beta_1(l, m)$ distribution, the random variable $Y = \frac{X}{1 - X}$ will indeed follow a $\beta_2(l, m)$ distribution.

Explanation:

• Beta Distribution (β1): The $\beta_1(l, m)$ distribution, commonly referred to simply as the Beta distribution, is defined for a random variable $X$ on the interval $(0, 1)$. The parameters $l$ and $m$ shape the distribution.

• Transformation: If $X$ has a $\beta_1(l, m)$ distribution, then the transformation $Y = \frac{X}{1-X}$ will map $X$ from the interval $(0, 1)$ to $(0, \infty)$.

• Beta Prime Distribution (β2): The distribution of $Y = \frac{X}{1-X}$ is known as the Beta prime distribution or the $\beta_2(l, m)$ distribution, which is defined on the interval $(0, \infty)$.

This relationship is due to the properties of the Beta distribution and how the transformation affects the probability density function.

Would you like further details or explanations on this topic?

Follow-up Questions:

1. What are the applications of the Beta prime distribution in statistics?
2. Can you show the derivation of the transformation $Y = \frac{X}{1-X}$ from $\beta_1(l, m)$ to $\beta_2(l, m)$?
3. How do the shape parameters $l$ and $m$ affect the distribution of $Y$?
4. What are some practical examples where the Beta prime distribution is used?
5. How does the Beta prime distribution differ from other common distributions like Gamma or Weibull?

Tip:

Understanding the transformation of variables and how distributions change under these transformations is crucial in advanced probability and statistics, particularly in Bayesian inference and reliability analysis.