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I am learning probability and have a (probably dumb) question. It's not a homework question, just something that confuses me.

If someone tells me that $X,Y\sim D$ where $D$ is some known distribution (e.g., Uniform, Normal, etc.) is it possible to say anything about the joint distribution $f_{X,Y}(x,y)$ of $X$ and $Y$ without knowing whether $X,Y$ are dependent?

For example, if a question reads

$X,Y\sim \text{Unif}(0,1),$ let $Z=\max{(X,Y)}$. Find $E(Z).$

is it possible to answer the question without assuming $X,Y$ are independent or determining the dependence relation between $X,Y$?

EssentialAnonymity
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No, given that $X, Y \sim D$, it is not necessarily the case that $X$ and $Y$ are independent. For your example, suppose that the joint density of $X$ and $Y$ has value $2$ on the square region $\left[0,\frac 12\right)\times \left[0,\frac 12\right)$ and also on the square region $\left[\frac 12,1\right]\times \left[\frac 12,1\right]$. Then, $X, Y \sim U[0,1]$ but they are not independent random variables. As another example, consider the case when $X, Y\sim N(0,1)$ but the joint density is $$f_{X,Y}(x,y) = \begin{cases} 2\phi(x)\phi(y), & \text{if}~ x \geq 0, y \geq 0, \text{or}~ x < 0, y < 0,\\ 0,& \text{otherwise},\end{cases}$$ where $\phi(x)$ is the standard normal pdf (cf. the latter part of this answer of mine).

Dilip Sarwate
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  • In your first example. if you didn't know the joint distribution pdf a priori, would it be possible to find $E(\max (X,Y))$? – EssentialAnonymity Oct 14 '20 at 16:20
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    Without the joint pdf, it is not possible to find $E[\max(X,Y)]$. My examples are just specific cases of what the joint pdf might be; there are infinitely many other possible joint pdfs which will result in different values for $E[\max(X,Y)]$. – Dilip Sarwate Oct 14 '20 at 16:24