4

If $X\sim U , Y\sim U$ , and $X,Y$ may be non-independent. Can we say the joint distribution of $X,Y$ is uniform?

kjetil b halvorsen
  • 77,844
aditer
  • 41

2 Answers2

6

No, the joint distribution is not necessarily uniform.

Consider $X$ and $Y$ with a joint pdf

$$ f(x,y) = \begin{cases} 2, \text{if } x \in (0,0.5), y \in (0,0.5)\\ 2, \text{if }x \in (0.5,1), y \in (0.5,1) \\ 0, \text{otherwise} \end{cases} $$

Then both $X$ and $Y$ have marginal $U(0,1)$ distributions, but the joint distribution is not uniform.

David Thiessen
  • 2,670
  • 2
    +1 I put together a visualization of your density. set.seed(2021); N <- 1000; x1 <- runif(N, 0, 0.5); x2 <- runif(N, 0.5, 1); y1 <- runif(N, 0, 0.5); y2 <- runif(N, 0.5, 1); x <- c(x1, x2); y <- c(y1, y2); par(mfrow = c(3, 1)); plot(x, y); plot(x, ecdf(x)(x), main = "CDF of x"); abline(a = 0, b = 1); plot(y, ecdf(y)(y), main = "CDF of y"); abline(a = 0, b = 1) – Dave Aug 09 '21 at 18:25
2

Suppose that $X\sim Unif(0,1)$ and $Y\sim Unif(0,X)$, then the joint distribution will be

$$f(X,Y)=f(Y|X)f(X)=\frac{1}{x}, \ \ \ 0\leq y \leq x\leq 1$$

If I'm not mistaken a uniform distribution gives the same importance to the whole domain of $(X,Y)$, whereas in the case displayed the importance given depends on the sampled $X$.

Fiodor1234
  • 2,224
  • 1
    I think it's worth noting that although $Y|X \sim Unif(0,X)$, the marginal distribution of $Y$ is not uniform. – David Thiessen Aug 10 '21 at 15:02
  • If the parameter of interval is a random variable, it is a uniform disturbution? In fact, many definitions do not clearly indicate this problem. Anyway, copula is a better interpretation to solve this question for me. – aditer Oct 08 '21 at 16:16