To prove that the transformation $Y = \frac{X}{1-X}$ leads from a Beta distribution $\beta_1(l, m)$ to a Beta prime distribution $\beta_2(l, m)$, we need to start by looking at the probability density functions (PDFs) of both distributions and perform the appropriate transformation.

1. Beta Distribution $\beta_1(l, m)$

The probability density function (PDF) of the Beta distribution $X \sim \beta_1(l, m)$ is given by:

$f_X(x) = \frac{x^{l-1}(1-x)^{m-1}}{B(l, m)}, \quad 0 < x < 1$

where $B(l, m)$ is the Beta function, defined as:

$B(l, m) = \int_0^1 t^{l-1} (1-t)^{m-1} \, dt$

2. Transformation to Beta Prime Distribution

Let $Y = \frac{X}{1-X}$. We need to find the distribution of $Y$.

First, solve for $X$ in terms of $Y$:

$X = \frac{Y}{1 + Y}$

Next, we find the derivative of $X$ with respect to $Y$:

$\frac{dX}{dY} = \frac{1}{(1+Y)^2}$

3. Change of Variables

To find the PDF of $Y$, we use the change of variables formula:

$f_Y(y) = f_X(x) \left| \frac{dx}{dy} \right|$

Substitute $x = \frac{y}{1+y}$ and $\frac{dx}{dy} = \frac{1}{(1+y)^2}$:

$f_Y(y) = f_X\left(\frac{y}{1+y}\right) \cdot \frac{1}{(1+y)^2}$

Substitute the expression for $f_X(x)$:

$f_Y(y) = \frac{\left(\frac{y}{1+y}\right)^{l-1}\left(1 - \frac{y}{1+y}\right)^{m-1}}{B(l, m)} \cdot \frac{1}{(1+y)^2}$

Simplify the expression inside the PDF:

$f_Y(y) = \frac{y^{l-1}(1+y)^{1-l} (1+y)^{1-m}}{B(l, m) (1+y)^2} = \frac{y^{l-1}}{B(l, m)} \cdot (1+y)^{-l-m}$

Thus, the PDF of $Y$ is:

$f_Y(y) = \frac{y^{l-1}}{B(l, m)} \cdot (1+y)^{-l-m}, \quad y > 0$

4. Beta Prime Distribution $\beta_2(l, m)$

The PDF of the Beta prime distribution $Y \sim \beta_2(l, m)$ is:

$f_Y(y) = \frac{y^{l-1}}{B(l, m)} (1+y)^{-(l+m)}, \quad y > 0$

Conclusion:

The derived PDF of $Y = \frac{X}{1-X}$ exactly matches the PDF of the Beta prime distribution $\beta_2(l, m)$. Therefore, we have proven that if $X$ follows a Beta distribution $\beta_1(l, m)$, then the transformed variable $Y = \frac{X}{1-X}$ follows a Beta prime distribution $\beta_2(l, m)$.

Would you like further explanations or another example?

Follow-up Questions:

1. How does the transformation impact the moments of the distribution?
2. Can you derive the cumulative distribution function (CDF) for the Beta prime distribution?
3. How would the proof change if we considered a different transformation, like $Y = \frac{1-X}{X}$?
4. What are the special cases of the Beta prime distribution for specific values of $l$ and $m$?
5. How is the Beta prime distribution used in real-world applications?

Tip:

When working with transformations of random variables, it's essential to carefully consider the domain and ensure that the new PDF is properly normalized across the transformed range.