Distribution of three dependent random variables I have three dependent random variables $X_1$, $X_2$, $X_3$, ($X_i\in\left[0, 1\right]$) that satisfy additional condition $\sum_i{X_i} = 1$.
If (marginal) distributions of $X_i$ ($i = 1, 2, 3$) are given, I need to determine likelihood of a certain combination of those variables. More formally, if $a_1, a_2, a_3\in \left[0, 1\right]$ such that $\sum_i{a_i} = 1$ and $\epsilon > 0$, I would like to know the probability:
$$
P((\forall i) |X_i - a_i| < \epsilon).
$$
 A: You can proceed some distance with that assumptions, but you cannot recover a specific expression for the probability because the joint distribution can be anything.  
Using the restriction on the rv's we have $X_3 =1- X_2-X_1$. Using the restrictions on the constants, we have $a_3=1-a_2-a_1$. Then we search for the joint probability
$$P( |X_1 - a_1| < \epsilon,\; |X_2 - a_2| < \epsilon,\;|X_3 - a_3| < \epsilon)=?$$ and substituting we obtain
$$P( |X_1 - a_1| < \epsilon,\; |X_2 - a_2| < \epsilon,\;|1- X_2-X_1 - (1-a_2-a_1)| < \epsilon)=?$$
$$P( |X_1 - a_1| < \epsilon,\; |X_2 - a_2| < \epsilon,\;|- (X_2-a_2)-(X_1 -a_1)| < \epsilon)=?$$
$$P( |X_1 - a_1| < \epsilon,\; |X_2 - a_2| < \epsilon,\;|(X_2-a_2)+(X_1 -a_1)| < \epsilon)=?$$
$$P( |X_1 - a_1| < \epsilon,\; |X_2 - a_2| < \epsilon,\;|(X_2+X_1)-(a_2+a_1)| < \epsilon)=?$$
We were able to take the third rv out of the picture, but the fact that you have available the marginal distributions does not lead you to a unique joint distribution: this is a well known result, i.e. that a set of given marginals are compatible with any number of joint distribution functions. What is the dependence sturcture? How $X_1$ and $X_2$ are related?
