The random variable V takes values in the set {0,1} and the random variable W takes values in the set {0,1,2}. Their joint PMF is of the form

pV,W(v,w)=c⋅(v+w),


where c is some constant, for v and w in their respective ranges, and is zero everywhere else.

a) Find the value of c.

c=?

b) Find pV(1).=?

a) To find the value of c, we need to consider all possible values of v and w in their respective ranges {0, 1} and {0, 1, 2}.

For v = 0 and w = 0, pV,W(0,0) = c⋅(0 + 0) = 0.
For v = 0 and w = 1, pV,W(0,1) = c⋅(0 + 1) = c.
For v = 1 and w = 0, pV,W(1,0) = c⋅(1 + 0) = c.
For v = 1 and w = 1, pV,W(1,1) = c⋅(1 + 1) = 2c.
For v = 0 and w = 2, pV,W(0,2) = c⋅(0 + 2) = 2c.
For v = 1 and w = 2, pV,W(1,2) = c⋅(1 + 2) = 3c.

Since the joint probability mass function (PMF) must sum to 1 over all possible values, we can equate the sum of all probabilities to 1:

0 + c + c + 2c + 2c + 3c = 1
9c = 1
c = 1/9

Therefore, the value of c is 1/9.

b) To find pV(1), we need to sum the joint probabilities for all values of w when v = 1:

pV(1) = pV,W(1,0) + pV,W(1,1) + pV,W(1,2)
= c + 2c + 3c
= 6c

Substituting the value of c we found earlier:

pV(1) = 6 * (1/9)
= 2/3

Therefore, pV(1) is equal to 2/3.