# Thread: Random vector distribution

1. ## Random vector distribution

Random variable has discrete distribution:
Conditional distribution of random vector under condition is:

a. Find distribution of random vector
b. Find conditional distribution under condition

How can I use Bayes' formula?
Thanks for any help.

2. For part (a) the unconditional distribution is the sum of the probabilities under all possible conditions
$f(u,v) = \sum_k P(X=k)f_{(U,V) | X=k}(u,v)$

For part (b) Baye's formula says
$P(A | B) = \frac{P(A,B)}{P(B)} = \frac{P(A)P(B|A)}{P(B)}$

which in this case becomes

$P(X=k | (u,v) = (U,V)) = \frac{P(X=k, (u,v) = (U,V))}{f(U,V)}
= \frac{P(X=k)f_{(U,V) | X=k}(u,v)}{f(U,V)}$