Besides jointly normal random variable, what other distribution satisfies uncorrelated if and only if independent?

Question

It is well known that for a jointly normally distributed random variables $(X_1,...,X_n)^T,$ they are uncorrelated if and only if independent.

It is also well-known that for any random variable, independent implies uncorrelated but not the converse.

Here comes my question:

Question: Besides jointly normal random variable, what other distribution satisfies uncorrelated if and only if independent?

Answer 1

As a trivial example:

Let $X,Y$ be two uncorrelated coin flips (i.e. each are $1$ with probability $1/2$ and $0$ with probability $1/2$ and $E(XY)=1/4$).

Then \begin{align*}\frac14 &=E(XY)\\&=1\cdot P(X=1,Y=1)+0\cdot P(X=1,Y=0)+0\cdot P(X=0,Y=1)+0\cdot P(X=0,y=0)\\&=P(X=1,Y=1)\end{align*}

Or $P(X=1,Y=1)=1/4=P(X=1)P(Y=1)$.

Then by law of total probability $P(X=1,Y=0)+P(X=1,Y=1)=1/2$ so $P(X=1,Y=0)=1/4=P(X=1)P(Y=0)$.

The rest follow.

Answer 2

Here we're talking about families of distributions rather than about individual distributions. If I specify just one distribution for a tuple $(X_1,\ldots,X_n)$ of random variables and it happens that they are independent and have finite variances, then I can say they're independent if and only if they're uncorrelated, and if it happens that they have all non-zero covariences, then I can also say they're independent if and only if they're uncorrelated. So any distribution matching either of these two descriptions is an example of the kind requested, but an uninteresting example.

Here's just one example that's somewhat more interesting. Suppose you have a Poisson process on a measure space (which, as often happens, could for example be the Euclidean plane with Lebesgue measure). Let $X_A$ be the number of "arrivals" within the measurable set $A,$ so that $X_A\sim\operatorname{Poisson}(m(A)),$ where $m(A)$ is the measure of $A.$ (In the plane example, one would usually just say that $m(A)$ is the area of the region $A$ in the plane.)

Then $X_{A_1}, \ldots, X_{A_n}$ are independent if and only if they are uncorrelated, and also if and only if $A_1,\ldots,A_n$ are pairwise disjoint.

One might say this is a family of distributions parametrized by the tuple $(A_1,\ldots,A_n),$ but the joint distribution depends on the choice of that tuple only through the measures of their intersections, so the latter should probably be taken to be the parameter space.