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Let $X$ and $Y$ be independent and identically distributed $(i.i.d.)$ r.v.’s, each having the probability distribution, $p(k) = (1 − λ)λ^k$; $k = 0,1,...$ where $λ :(0; 1)$ is a constant. Define $U = min(X; Y )$; $V = max(X; Y )$; $W = V − U$. Determine the joint probability distribution of $U$ and $W$ (taking care with $W = 0$) and verify that $U$ and $W$ are independent r.v.’s.

My work: I set up this: $P(X=W+U, Y=U)$ when $X>Y$ and $P(X=U, Y=W+U)$ when $X<Y$. and the joint distributions are the same in both cases as $W$ is always non-negative. finally I got the following joint pmf: $f(w,u)=(1-λ)^2 λ^(w+2u-2)$ is this the correct joint pmf?

Let $X$ and $Y$ be independent and identically distributed $(i.i.d.)$ r.v.’s, each having the probability distribution, $p(k) = (1 − λ)λ^k$; $k = 0,1,...$ where $λ :(0; 1)$ is a constant. Define $U = min(X; Y )$; $V = max(X; Y )$; $W = V − U$. Determine the joint probability distribution of $U$ and $W$ (taking care with $W = 0$) and verify that $U$ and $W$ are independent r.v.’s.

My work: I set up this: $P(X=W+U, Y=U)$ when $X>Y$ and $P(X=U, Y=W+U)$ when $X<Y$. and the joint distributions are the same in both cases as $W$ is always non-negative. finally I got the following joint pmf: $f(w,u)=(1-λ)^2 λ^(w+2u-2)$ when $X=Y$, $f(w,u)=(1-λ)^2 λ^(2u-2)$ and ${(w,u): w=0,1...; u= 0,1,..}$ is this the correct joint pmf? what will be the final joint pmf?

Let $X$ and $Y$ be independent and identically distributed $(i.i.d.)$ r.v.’s, each having the probability distribution, $p(k) = (1 − λ)λ^k$; $k = 0,1,...$ where $λ :(0; 1)$ is a constant. Define $U = min(X; Y )$; $V = max(X; Y )$; $W = V − U$. Determine the joint probability distribution of $U$ and $W$ (taking care with $W = 0$) and verify that $U$ and $W$ are independent r.v.’s.

Let $X$ and $Y$ be independent and identically distributed $(i.i.d.)$ r.v.’s, each having the probability distribution, $p(k) = (1 − λ)λ^k$; $k = 0,1,...$ where $λ :(0; 1)$ is a constant. Define $U = min(X; Y )$; $V = max(X; Y )$; $W = V − U$. Determine the joint probability distribution of $U$ and $W$ (taking care with $W = 0$) and verify that $U$ and $W$ are independent r.v.’s.

My work: I set up this: $P(X=W+U, Y=U)$ when $X>Y$ and $P(X=U, Y=W+U)$ when $X<Y$. and the joint distributions are the same in both cases as $W$ is always non-negative. finally I got the following joint pmf: $f(w,u)=(1-λ)^2 λ^(w+2u-2)$ is this the correct joint pmf?