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.
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.
For part (a) the unconditional distribution is the sum of the probabilities under all possible conditions
$\displaystyle f(u,v) = \sum_k P(X=k)f_{(U,V) | X=k}(u,v)$
For part (b) Baye's formula says
$\displaystyle P(A | B) = \frac{P(A,B)}{P(B)} = \frac{P(A)P(B|A)}{P(B)}$
which in this case becomes
$\displaystyle P(X=k | (u,v) = (U,V)) = \frac{P(X=k, (u,v) = (U,V))}{f(U,V)}
= \frac{P(X=k)f_{(U,V) | X=k}(u,v)}{f(U,V)}$