Let X and Y be independent exponential random variables with...

Let $X$ and $Y$ be independent exponential random variables with respective rates $\lambda$ and $\mu$. (a) Argue that, conditional on $X>Y$, the random variables $\min (X, Y)$ and $X-Y$ are independent. (b) Use part (a) to conclude that for any positive constant $c$ $$ \begin{aligned} E[\min (X, Y) \mid X>Y+c] &=E[\min (X, Y) \mid X>Y] \\ &=E[\min (X, Y)]=\frac{1}{\lambda+\mu} \end{aligned} $$ (c) Give a verbal explanation of why $\min (X, Y)$ and $X-Y$ are (unconditionally) independent.

Let $X$ and $Y$ be independent exponential random variables with respective rates $\lambda$ and $\mu$.
(a) Argue that, conditional on $X>Y$, the random variables $\min (X, Y)$ and $X-Y$ are independent.
(b) Use part (a) to conclude that for any positive constant $c$
$$
\begin{aligned}
E[\min (X, Y) \mid X>Y+c] &=E[\min (X, Y) \mid X>Y] \\
&=E[\min (X, Y)]=\frac{1}{\lambda+\mu}
\end{aligned}
$$
(c) Give a verbal explanation of why $\min (X, Y)$ and $X-Y$ are (unconditionally) independent.

Key Concepts

Order Statistics and the Independence of Gaps

Order statistics involve the sorted values of a set of random variables, with the minimum being the first order statistic. In the context of exponential distributions, the gap between order statistics (for instance, the difference between two ordered observations) can become independent of the minimum value due to the memoryless property. This concept explains why, under certain conditions or even unconditionally, function transformations such as the minimum and the gap may exhibit independent behavior.

Conditional Independence in Statistical Transformations

Conditional independence refers to a situation where two random variables become independent once a conditioning event is imposed. In probabilistic models, particularly when dealing with functions of random variables like the minimum and the difference, establishing conditional independence can simplify the analysis. It allows the separate treatment of components that are otherwise intertwined in their joint distribution.

Transformation of Random Variables

Transforming random variables by considering functions such as the minimum or the difference is a common technique in probability theory. This approach allows one to re-express joint distributions in a way that can reveal independence or simplify the computation of statistical properties. The transformation often leverages the inherent properties of the original distributions, such as the memoryless characteristic of exponential distributions.

Exponential Distribution and Memoryless Property

The exponential distribution is characterized by a constant hazard rate and the unique memoryless property. This property means that the probability of an event occurring in the future does not depend on the past and is a crucial aspect when analyzing waiting times and event occurrences. Its use simplifies the analysis of interarrival times and supports conclusions about the independence of time-related transformations.

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