The random variable X has a range of {0,1,2} and the random variable Y has a range of {1,2}.
The joint distribution of X and Y is given by the following table:
x y Ρ = = ( X x Y y , )
0 1 0.2
0 2 0.1
1 1 0.0
1 2 0.2
2 1 0.3
2 2 0.2
To find the marginal distribution of X and Y, we sum across the rows for X and across the columns for Y:
Marginal distribution of X:
P(X=0) = P(X=0, Y=1) + P(X=0, Y=2) = 0.2 + 0.1 = 0.3
P(X=1) = P(X=1, Y=1) + P(X=1, Y=2) = 0.0 + 0.2 = 0.2
P(X=2) = P(X=2, Y=1) + P(X=2, Y=2) = 0.3 + 0.2 = 0.5
Marginal distribution of Y:
P(Y=1) = P(X=0, Y=1) + P(X=1, Y=1) + P(X=2, Y=1) = 0.2 + 0.0 + 0.3 = 0.5
P(Y=2) = P(X=0, Y=2) + P(X=1, Y=2) + P(X=2, Y=2) = 0.1 + 0.2 + 0.2 = 0.5
Therefore, the marginal distribution of X is {0.3, 0.2, 0.5} and the marginal distribution of Y is {0.5, 0.5}.