As someone who is trying to pick probability and statistics back up after not using it for the last 4 years, I'd like to ask for some help with a question I encountered on a MIT OCW midterm test (see below for links).
I have been given independent exponential random variables $X \sim Exp(\lambda_1 = 1)$, $Y \sim Exp(\lambda_2 = 2)$, $Z \sim Exp(\lambda_3 = 3)$ with $M = \max\{X, Y, Z\}$ and $L = \min\{X, Y, Z\}$, and told that $L$ and $M - L$ are independent.
The only line referring to this in the partial solutions (see link below) states that the "Idea is to argue first that $\min\{X,Y,Z\}$ and $\max\{X,Y,Z\}-\min\{X,Y,Z\}$ are independent" but I would like to know how someone would argue or show this.
I've considered an approach where I try to show the joint distribution is the product of the marginal distributions but I'm not confident my working is correct.
The promptness of the solution description makes me think there must be a different way to approach or think about this problem and having looked around, I've noticed there are a few threads on the distribution of $M-L$ if you have 2 i.i.d. exponential RVs but not for multiple exponential RVs with different $\lambda_i$.
My thoughts going into this:
If RVs are independent, then their joint distribution is the product of their marginal distributions: $f(x_1,...,x_n) = \Pi_{i=1}^{n}f_i(x_i)$ (and similarly with CDFs). Additionally, covariance = 0. If I can show any of these, then I can verify the statement.
Exponential RVs can be interpreted as "time elapsed before an event occurs" and have the memoryless property: $P(T > t_1 + t_2 | T > t_1) = P(T > t_2)$. Is it worth considering the sum of these elapsed times as ways to represent $M$ and $L$? $L\sim Exp(\lambda = \lambda_1 + \lambda_2 + \lambda_3)$ but $M$ is not an Exponential RV. I would also have to consider the different possible orders of $X,Y,Z$ when summing the times up to be equivalent to $M$.
You can derive a joint distribution from a conditional and marginal distribution, $f(x, y) = f_{X|Y}(x)f_Y(y)$, and I can derive $f_{M-L}(a) = \int f_M(m)f_L(m - a)dm$.
Information calculated:
- $F_M(a) = P(\max\{X,Y,Z\} \leq a) = (1- e^{-\lambda_1 a})(1- e^{-\lambda_2 a})(1- e^{-\lambda_3 a})$
- $f_M(m) = \frac{d}{da}F_M(a)$
- $F_L(a) = 1 - P(\min\{X,Y,Z\} \geq a) = 1 - e^{-(\lambda_1 + \lambda_2 + \lambda_3)a}$
- $f_L(l) = \lambda e^{-\lambda l}$ where $\lambda = \lambda_1 + \lambda_2 + \lambda_3$
What I attempted:
I considered trying to verify independence by showing $f(x_1,...,x_n) = \Pi_{i=1}^{n}f_i(x_i)$. I think I can compute $f_{M-L}(a) = \int f_M(m)f_L(m-a) dm$ or $F_{M-L}(a)$. I'm not given the joint distribution but I think I could try to aim for it using conditional probability relation: $$f_{M-L|L=l}(a) = \frac{f_{M-L,L}(a, l)}{f_L(l)}$$ If I consider $F_{M-L}(a) = P(M-L \leq a)$, then I also have that $$F_{M-L|L}(a) = P(M-L \leq a | L = l)$$ Where I think $$P(M-L \leq a | L = l) = P(M \leq a + L | L = l)$$ $$\int_0^a f_{M-L|L}(r)dr = \int_0^{a+l} f_{M | L} (m) dm$$ And if I let $m = \frac{a+l}{a}r$: $$\int_0^{a} f_{M-L|L}(r)dr = \int_0^{a} f_{M | L} \bigg(\frac{a+l}{a}r\bigg) \frac{a+l}{a}dr$$ $$\implies f_{M-L|L}(r) = f_{M | L} \bigg(\frac{a+l}{a}r\bigg) \frac{a+l}{a} = \frac{f_{M-L,L}(r,l)}{f_L(l)}$$ And so if $f_{M | L} \bigg(\frac{a+l}{a}r\bigg) \frac{a+l}{a} = f_{M-L}(r)$ then $M-L$ and $L$ are independent.
I'm uncertain about this because now I have introduced some arbitrary constant $a$ and I'm not certain how to proceed regarding calculating $f_{M|L}(m)$ since I haven't shown if $M$ and $L$ are independent of each other so I don't want to say $f_{M, L}(m,l) = f_M(m)f_L(l)\}$.
I'd be grateful for any insight into how to approach determining if $M-L$ and $L$ are independent as well as any pointers as to whether my approach is correct and if I have any errors or misconceptions.
Thank you very much for your help!
MIT OCW course followed: https://ocw.mit.edu/courses/18-440-probability-and-random-variables-spring-2014/
Query based off question 5(e) from: https://ocw.mit.edu/courses/18-440-probability-and-random-variables-spring-2014/c7f788511ef3812ac7b3d60af57032ae_MIT18_440S14_prctcmdtrm2.pdf
Provided partial solution: https://ocw.mit.edu/courses/18-440-probability-and-random-variables-spring-2014/128cda4cdf29684293f78b56d0a08627_MIT18_440S14_prctcmdtrm2sl.pdf