A fair uniform spinner is spun twice, and the results V and W are noted. V and W are uniform RVs ∼U[0,1].
I'm trying to answer the question what is the joint pdf for V and W. I know that I have to integrate from 0 to 1 twice, once for each V and W respectively, but stuck on figuring out what I am integrating. I know it should be f(x,y) but how do I find that?
Thanks.
You have: $ U, W : U\sim\mathcal{U}(0,1), W\sim\mathcal{U}(0,1), U\bot W$
That is, two independent and identically distributed (iid) random variables each with uniform distribution over the range $[0,1]$.
For Uniform Distribution over the range $[a,b]$: $$X\sim\mathcal{U}(a, b) \iff \Pr(X\leq x) = \begin{cases}0 & x\lt a \\ \int\limits_a^x \frac{1}{b-a}\;\mathrm{d}\chi & a\leq x \leq b \\ 1 & x\gt b\end{cases}$$
$\text{i.e. } \underbrace{ f_X(x)= \begin{cases}\frac{1}{b-a} & a\leq x \leq b \\ 0 & \text{elsewhere}\end{cases} }_{\text{probability density function (pdf)}},\quad \underbrace{\operatorname{F}_X(x)=\begin{cases}0 & x\lt a \\ \frac{x-a}{b-a} & a\leq x \leq b \\ 1 & x> b\end{cases}}_{\text{cumulative distribution function (CDF)}}$
For independent continuous random variables: $X\bot Y \iff f_{X,Y}(x,y)=f_X(x)\cdot f_Y(y)$
$F_{X,Y}(x,y) = \int\limits_{-\infty}^{x}\,\int\limits_{-\infty}^{y} f_{X,Y}(\chi,\psi)\;\mathrm{d}\chi\,\mathrm{d}\psi$
Put this together to find $f_{U,W}(u,w)$ and hence $F_{U,W}(u, w) \color{grey}{= \Pr(U\leq u\cap W\leq w)}$