Let $X=(X\_1,\dots,X\_d)$ be a random vector, and a.s. $X \in [0,1]^d$. Suppose that for every $a \in \mathbb{R}^d$, we know the probability distribution of the random variable $Y\_a = <a,X>$. My question is that can one uniquely determine the distribution of $X$ (i.e., the joint distribution of $X\_1,\dots,X\_d$)?


If the answer of the above question is *yes*, here is a further question. Are there a positive integer $n$, and $a\_1,\dots,a\_n$ ($a\_i \in \mathbb{R}^d$) so that if we know the distributions of $Y\_{a\_1},\dots,Y\_{a\_n}$, where $Y\_{a\_i}=<a\_i, X>$, then the distribution of $X$ can be uniquely determined?



Determine joint distribution from projections
The answer to the first question is yes. Think of the characteristic function of X, namely
$$
 \varphi\_X(a)
 = \mathbb{E}[e^{i\langle a,X\rangle}]
 = \mathbb{E}[e^{i Y\_a}]
$$
For the second question, I would say the answer is no. If you know $(Y\_{a\_i})\_{i=1,\dots,n}$, it amounts to know $\varphi\_X$ along the directions $a\_1,\dots,a\_n$. Since the characteristic function is no more than a Fourier transform, you may look at the problem from an analytic point of view: is the Fourier transform determined by the values on a finite number of directions? I think it is now easy to find counterexamples.


