What is the value of c in the joint PMF equation pV,W(v,w)=c⋅(v+w)?
To find the value of c in the joint PMF equation pV,W(v,w) = c⋅(v + w), we need to use the properties of joint probability mass functions (PMFs). The joint PMF assigns a probability value to each pair of random variables (v, w) in a given probability distribution.
To determine c, we must ensure that the sum of all the probabilities in the joint PMF equals 1. In other words, the joint PMF must be a valid probability distribution.
Since pV,W(v, w) = c⋅(v + w), we can calculate the sum of the probabilities as follows:
sum(pV,W(v, w)) = sum(c⋅(v + w)).
To evaluate this sum, we need to determine the range of possible values for v and w. Once we know these ranges, we can calculate the sum over all valid pairs of (v, w).
Given this information, we can divide both sides of the equation by c to isolate (v + w):
sum((v + w)) = sum((pV,W(v, w)) / c).
If we evaluate the sum on the left side and divide the sum((pV,W(v, w)) / c) by c, we should obtain the sum of probabilities over all possible pairs of (v, w).
Finally, we set this sum equal to 1 and solve for c:
sum((pV,W(v, w)) / c) = 1.
By rearranging the equation, we can solve for c:
c = sum(pV,W(v, w)) / sum((v + w)).
Calculating the sums in the equation above will give us the value of c in the joint PMF equation pV,W(v,w) = c⋅(v + w).