Suppose that Y1 and Y2 are independent exponentially...

Suppose that $Y_{1}$ and $Y_{2}$ are independent exponentially distributed random variables, both with mean $\beta$, and define $U_{1}=Y_{1}+Y_{2}$ and $U_{2}=Y_{1} / Y_{2}.$ a. Show that the joint density of $\left(U_{1}, U_{2}\right)$ is $$ f_{U_{1}, U_{2}}\left(u_{1}, u_{2}\right)=\left\{\begin{array}{ll} \frac{1}{\beta^{2}} u_{1} e^{-u_{1} / \beta} \frac{1}{\left(1+u_{2}\right)^{2}}, & 0<u_{1}, 0<u_{2} \\ 0, & \text { otherwise } \end{array}\right. $$ b. Are $U_{1}$ and $U_{2}$ are independent? Why?

Suppose that $Y_{1}$ and $Y_{2}$ are independent exponentially distributed random variables, both with mean $\beta$, and define $U_{1}=Y_{1}+Y_{2}$ and $U_{2}=Y_{1} / Y_{2}.$
a. Show that the joint density of $\left(U_{1}, U_{2}\right)$ is
$$
f_{U_{1}, U_{2}}\left(u_{1}, u_{2}\right)=\left\{\begin{array}{ll}
\frac{1}{\beta^{2}} u_{1} e^{-u_{1} / \beta} \frac{1}{\left(1+u_{2}\right)^{2}}, & 0<u_{1}, 0<u_{2} \\
0, & \text { otherwise }
\end{array}\right.
$$
b. Are $U_{1}$ and $U_{2}$ are independent? Why?

Key Concepts

Exponential Distribution

The exponential distribution is a continuous probability distribution characterized by its constant hazard rate and memoryless property. It is typically used to model the time between events in a Poisson process. In this context, the fact that the variables are exponentially distributed with the same mean is central to computing the joint density after applying the transformation.

Transformation of Random Variables

When new random variables are defined as functions of existing ones, the probability distributions must be transformed accordingly. This process involves expressing the joint density of the new variables in terms of the original random variables and applying the proper change-of-variable techniques.

Jacobian Determinant

The Jacobian determinant is a key tool in multivariable calculus used to account for scaling effects when variables are transformed. In the context of probability, it is essential to correct for the change in volume element when moving from the original variable space to the new one.

Joint Density Functions

The joint density function provides a complete description of the distribution of two or more random variables simultaneously. It allows for the analysis of relationships between the variables, including whether they are statistically independent or not.

Independence of Random Variables

Independence in probability is defined by the factorization of the joint density into the product of marginal densities. In the context of transformed variables, determining independence involves checking whether such a factorization holds. The example investigates this property for the sum and the ratio of independent exponential variables.

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