# Bivariate Distribution with Uniform Marginals is Bound to be Uniform?

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?

• A term that might interest you is copula.
– Dave
Aug 9, 2021 at 18:05
• Consider the case $X=Y.$
– whuber
Aug 9, 2021 at 21:32
• See Is there a bivariate β distribution I can fit to my data? for examples! (choose the parameters so that the marginals are uniform). Aug 10, 2021 at 0:43
• Consider a chessboard, and imagine the probability is uniform on the white squares Aug 10, 2021 at 1:29

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.

• +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 9, 2021 at 18:25

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 $$X$$.

• I think it's worth noting that although $Y|X \sim Unif(0,X)$, the marginal distribution of $Y$ is not uniform. Aug 10, 2021 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. Oct 8, 2021 at 16:16