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

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.

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.

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.

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.

Transcript

00:01 We're given independent exponential random variables x and y with a common parameter of lambda.

00:10 In part a, we're asked to use convolution to show the x plus y as a gamma distribution and to find the parameters of that distribution.

00:23 Well, we have the probability density function of x because it is an exponential distribution is lambda times e to the negative lambda x, where x is greater than 0.

00:50 And likewise, this is also the probability density function of y.

00:59 Lambda e to the negative lambda y, or y is greater than 0.

01:04 So the probability density function of the variable w, which will take to vx plus y, be determined using convolution, probability density function.

01:26 This is the convolution, the probability density function for x with the probability density function for y, which is the integral from negative infinity to infinity of fx.

01:46 Of x times f y of instead of y we have w minus x which is equal to y according to our random variables vx in substituting we get integral from negative infinity to infinity of really we're only considering x and y greater than zero so x and y are both greater than zero this implies that x is greater than 0, w minus x is also greater than 0, so we have the x is less than w, and therefore the x lies between 0 and w.

02:47 So we're really only integrating instead from negative infinity to infinity from 0 to w.

02:58 This is the space on which both of these are non -zero.

03:02 And we get lambda e to the negative lambda x times the lambda e to the negative lambda w minus x dx.

03:22 And this can be written as the integral from 0 to w of lambda squared.

03:34 And then we have e the negative lambda x times e to the lambda x becomes e to the 0, which is 1, and then times e to the negative lambda w d x.

03:56 And so we can clearly just factor out the lambda squared e to the negative, e to the negative lambda w and integrates to get lambda squared w e to the negative lambda w for w greater than zero.

04:29 Now we see that this is a gamma distribution with parameters alpha equaling 2 in the parameter beta equaling 1 over lambda...

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