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?