Hey guys, in this video we're going to do this statistics and probability problem.
This is a joint probability question, so we're given a joint PMF and we're asked
to find one probability, which is the probability that X is less than Y and
we're also asked to find two marginal probabilities and we're given the values
of X and Y. So this is a PMF joint probability distribution, so it's
discrete, which is an easier case than if it were a PDF, which would be a
continuous case. Let's go ahead and get started. For A, we're asked to find
the probability that X is less than Y and since we're given all the values of
X and Y, this is just going to be basically plugging in the values that
hold for X is less than Y. That's going to be 1 is less than 2 and the
probability that 1 is less than 3 plus the probability that 1 is less than 4.
So Y goes up to 4. And then for X is equal to 2, that's going to be 2 less than 3
plus 2 less than 4 and to get the actual probabilities we just plug in X and Y
for each probability into the PMF. It's very easy to find a probability
given the PMF for a discrete case, so let's go ahead and plug that in. We're
given that the PMF is X plus Y over 32. That's going to be 1 plus 2 over 32
plus 1 plus 3 over 32 plus 1 plus 4 plus 2 plus 3 plus 2 plus 4. And these are all
over 32. And if we add that up that's going to give us 23 over 32 is the
probability that X is less than Y. That's the solution for A, for B we're
asked to find the marginal PMF for X, so Px of X. The formula for the discrete
case for finding Px of X is equal to the summation of the joint over all of the
possible values of Y. For this, the marginal
is going to depend on the values of X. We're going to have a different marginal
for when X is equal to 1 and when X is equal to 2
and then we just plug in ones and then all the values of Y and 2 and all the
values of Y again to get the marginals-- the two marginal values of the Xs.
When X is equal to one, we're going to have 1 plus 1 plus 1 plus 2 plus
1 plus 3 plus 1 plus 4 over 32 and that gives us 14 over 32. And when
X is equal to 2, we're going to have 2 plus 1 plus 2 plus 2 plus 2
plus 3 plus 2 plus 4 over 32 and that's going to give us 18 over 32.
This is our answer for B and we do a similar calculation for the marginal
probability of the Ys. Py of Y is the marginal summation over all of the Xs
of the joint PMF. For this problem, that means that the marginal PMF is
going to be different depending on when Y equals 1, 2, 3, and 4 and when
Y equals 1 we're going to have 1 plus 1 plus 1 plus 2 over 32 and
that gives us 5 over 32. When Y is equal to 2 we're going to have 2
plus 1 plus 2 plus 2 over 32 and that gives us 7 over 32. When Y is
equal to 3, we're going to have 3 plus 1 plus 3 plus 2. That gives
us 9 over 32. And when Y is equal to 4, we're going to have 4 plus 1
plus 4 plus 2 and that gives us 11 over 32. So that's our answer for C, and we're
done with this problem :)
*chiptune music*