Summary of basic probability theory, part 3
D. Joyce, Clark University
Math 218, Mathematical Statistics, Jan 2008
Joint distributions. When studing two related
real random variables Xand Y, it is not enough
just to know the distributions of each. Rather, the
pair (X, Y ) has a joint distribution. You can think
of (X, Y ) as naming a single random variable that
takes values in the plane R2.
Joint and marginal probability mass func-
tions. Let’s consider the discrete case first where
both Xand Yare discrete random variables.
The probability mass function for Xis fX(x) =
P(X=x), and the p.m.f. for Yis fY(y) = P(Y=y).
The joint random variable (X, Y ) has its own p.m.f.
denoted f(X,Y )(x, y), or more briefly f(x, y ):
f(x, y) = P((X, Y )=(x, y)) = P(X=xand Y=y),
and it determines the two individual p.m.f.s by
fX(x) = X
y
f(x, y), fY(y) = X
x
f(x, y).
The individual p.m.f.s are usually called marginal
probability mass functions.
For example, sssume that the random variables
Xand Yhave the joint probability mass function
given in this table.
Y
−1 0 1 2
−1 0 1/36 1/6 1/12
X0 1/18 0 1/18 0
1 0 1/36 1/6 1/12
2 1/12 0 1/12 1/6
By adding the entries row by row, we find the the
marginal function for X, and by adding the entries
column by column, we find the marginal function
for Y. We can write these marginal functions on
the margins of the table.
Y fX
−1 0 1 2
−1 0 1/36 1/6 1/12 5/18
X0 1/18 0 1/18 0 1/9
1 0 1/36 1/6 1/12 5/18
2 1/12 0 1/12 1/6 1/3
fY5/36 1/18 17/36 1/3
Discrete random variables Xand Yare indepen-
dent if and only if the joint p.m.f is the product of
the marginal p.m.f.s
f(x, y) = fX(x)fY(y).
In the example above, Xand Yaren’t independent.
Joint and marginal cumulative distribu-
tion functions. Besides the p.m.f.s, there are
joint and marginal cumulative distribution func-
tions. The c.d.f. for Xis FX(x) = P(X≤x), while
the c.d.f. for Yis FY(y) = P(Y≤y). The joint
random variable (X, Y ) has its own c.d.f. denoted
F(X,Y )(x, y), or more briefly F(x, y ):
F(x, y) = P(X≤xand Y≤y),
and it determines the two marginal p.m.f.s by
FX(x) = lim
y→∞ F(x, y), FY(y) = lim
x→∞ F(x, y).
Joint and marginal probability density
functions. Now let’s consider the continuous case
where Xand Yare both continuous. The last
paragraph on c.d.f.s still applies, but we’ll have
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