In Example 6.14 . Y 1 and Y 2 were independent exponentially...

In Example $6.14 . \mathrm{Y} 1$ and $\mathrm{Y} 2$ were independent exponentially distributed random variables, both with mean $\beta$. We defined $U 1=Y 1 /(Y 1+Y 2)$ and $U 2=Y 1+Y 2$ and determined the joint density of $(\mathrm{U} 1, \mathrm{U} 2)$ to be $f \cup 1, U 2(u 1, u 2)=\{1 \beta 2 u 2 e-u 2 / \beta, 0<u 1<1,0<u 2,0,$ otherwise a. Show that $U$ 1 is uniformly distributed over the interval (0,1) b. Show that U 2 has a gamma density with parameters $\alpha=2$ and $\beta$. c. Establish that U 1 and U 2 are independent.

In Example $6.14 . \mathrm{Y} 1$ and $\mathrm{Y} 2$ were independent exponentially distributed random variables, both with mean $\beta$. We defined $U 1=Y 1 /(Y 1+Y 2)$ and $U 2=Y 1+Y 2$ and determined the joint density of $(\mathrm{U} 1, \mathrm{U} 2)$ to be
$f \cup 1, U 2(u 1, u 2)=\{1 \beta 2 u 2 e-u 2 / \beta, 0<u 1<1,0<u 2,0,$ otherwise
a. Show that $U$ 1 is uniformly distributed over the interval (0,1)
b. Show that U 2 has a gamma density with parameters $\alpha=2$ and $\beta$.
c. Establish that U 1 and U 2 are independent.

Key Concepts

Transformation of Random Variables

This concept involves converting one set of random variables into another set through a functional relationship. The change-of-variables method is used to derive the joint density of the new variables from the original density, which is fundamental when analyzing derived distributions.

Jacobian Determinant

The Jacobian determinant arises in the change-of-variables formula as the absolute value of the determinant of the partial derivatives matrix. It adjusts the density appropriately under the transformation, ensuring that the total probability remains one, and it is crucial in correctly deriving the new joint density.

Marginal Distributions

Marginalization is the process of integrating a joint probability density over one or more variables to obtain the density function of a subset of variables. This technique is key for understanding the behavior of individual variables derived from a joint distribution.

Gamma Distribution

The Gamma distribution is a continuous probability distribution defined by a shape parameter and a scale parameter. It is often used to model sums of exponential random variables and waiting times, and is characterized by its flexible shape that can model a wide range of behaviors in stochastic processes.

Uniform Distribution

A Uniform distribution, especially over the interval (0,1), is characterized by a constant density, meaning that every outcome within the interval is equally likely. This basic yet important distribution serves as a building block in many probability and simulation problems.

Independence of Random Variables

The concept of independence in random variables means that the joint probability density function can be factored into the product of the individual marginal densities. This property simplifies analysis and is fundamental in both theoretical and applied probability, as it indicates that the behavior of one variable does not affect the other.

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