I am stuck with a problem at work. Can anybody please help me to give me the joint distribution of $n$ Bernoulli random variables but under the constraint that the sum of the these $n$ random variables must be $1$.
Can anybody show me how to derive this distribution?
This is a categorical distribution, also known as a multinomial distribution with number of trials equal to $1$.
If the binomial probabilities are $q_k, \;k=1,\dots n$ then the multinomial probability is $$p_k=\frac {q_k\prod_{j\neq k} (1-q_j)}{\sum_r q_r\prod_{s\neq r}(1-q_s)}$$
To derive this you simply use the conditional probability $p (A_k|B)=p (A_kB)/p (B) $ where $A_k$ is the event "variable $k $ is equal to $1$" and $B $ is the event "sum of all $n $ variables equals 1". Then you can deduce that for both $A_k $ and $B $ to be true, all the other bernoulli variables must be zero. This probability is the numerator for the value of $p_k $ I gave earlier. Then $p (B)=\sum_r p (A_rB)$ using law of total probability and independence and the denominator I gave.
There are only $n$ ways the variables can sum to $1$: one of them will equal $1$ and the other $n-1$ will equal zero. The very phrasing of the question indicates the variables are exchangeable: thus, the joint distribution will not change when the variables are permuted. Since permutations of the variables merely change each of these outcomes into other outcomes, they are all equally likely. Consequently the distribution is the uniform one on those $n$ outcomes, with probability $1/n$ for each outcome. That fully describes the joint distribution.
The original question assumed neither exchangeability nor independence. But without making some such assumption, the only conclusion we can draw is that the joint distribution is some distribution on the $n$ possible outcomes I have described. The probabilities may be any $n$ non-negative values that sum to unity, as required by the axioms of probability.
