From Algorithms to Z-Scores - matloff - University of California, Davis

150 CHAPTER 8. MULTIVARIATE PMFS AND DENSITIES
a random variable X, defined as pX(i) = P (X = i), where i ranged over all values that X takes on.
We do the same thing for a pair of random variables:
Definition 20 For discrete random variables U and V, their probability mass function is defined
to be
pU,V (i, j) = P (U = i and V = j) (8.2)
where (i,j) ranges over all values taken on by (U,V). Higher-dimensional pmfs are defined similarly,
e.g.
pU,V,W (i, j, k) = P (U = i and V = j and W = k) (8.3)
So in our marble example above, pY,B(1, 2) = 0.048, pY,B(2, 0) = 0.012 and so on.
Just as in the case of a single discrete random variable X we have
P (X ∈ A) =
pX(i) (8.4)
for any subset A of the range of X, for a discrete pair (U,V) and any subset A of the pair’s range,
we have
i∈A
P [(U, V ) ∈ A) = pU,V (i, j) (8.5)
(i,j)∈A
Again, consider our marble example. Suppose we want to find P (Y < B). Doing this “by hand,”
we would simply sum the relevant probabilities in the table above, which are marked in bold face
below:
i ↓, j → 0 1 2 3
0 0.002 0.024 0.036 0.008
1 0.162 0.073 0.048 0.004
2 0.012 0.024 0.006 0.000
The desired probability would then be 0.024+0.036+0.008+0.048+0.004 = 0.12.
Writing it in the more formal way using (8.5), we would set
A = {(i, j) : i < j} (8.6)