Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

2.4. Independent Random Variables 121

Instead of working with pdfs (or pmfs) we could have presented independence
in terms of cumulative distribution functions. The following theorem shows the
equivalence.

Theorem 2.4.2.Let(X 1 ,X 2 )have the joint cdfF(x 1 ,x 2 )and letX 1 andX 2 have
the marginal cdfsF 1 (x 1 )andF 2 (x 2 ), respectively. ThenX 1 andX 2 are independent
if and only if

F(x 1 ,x 2 )=F 1 (x 1 )F 2 (x 2 ) for all(x 1 ,x 2 )∈R^2. (2.4.1)

Proof:We give the proof for the continuous case. Suppose expression (2.4.1) holds.

Then the mixed second partial is

∂^2

∂x 1 ∂x 2

F(x 1 ,x 2 )=f 1 (x 1 )f 2 (x 2 ).

Hence,X 1 andX 2 are independent. Conversely, supposeX 1 andX 2 are indepen-
dent. Then by the definition of the joint cdf,

F(x 1 ,x 2 )=

∫x 1

−∞

∫x 2

−∞

f 1 (w 1 )f 2 (w 2 )dw 2 dw 1

=

∫x 1

−∞

f 1 (w 1 )dw 1 ·

∫x 2

−∞

f 2 (w 2 )dw 2 =F 1 (x 1 )F 2 (x 2 ).

Hence, condition (2.4.1) is true.

We now give a theorem that frequently simplifies the calculations of probabilities
of events that involves independent variables.

Theorem 2.4.3.The random variablesX 1 andX 2 are independent random vari-
ables if and only if the following condition holds,

P(a<X 1 ≤b, c < X 2 ≤d)=P(a<X 1 ≤b)P(c<X 2 ≤d) (2.4.2)

for everya<bandc<d,wherea, b, c,anddare constants.

Proof:IfX 1 andX 2 are independent, then an application of the last theorem and
expression (2.1.2) shows that

P(a<X 1 ≤b, c < X 2 ≤d)=F(b, d)−F(a, d)−F(b, c)+F(a, c)

= F 1 (b)F 2 (d)−F 1 (a)F 2 (d)−F 1 (b)F 2 (c)

+F 1 (a)F 2 (c)

=[F 1 (b)−F 1 (a)][F 2 (d)−F 2 (c)],

which is the right side of expression (2.4.2). Conversely, condition (2.4.2) implies
that the joint cdf of (X 1 ,X 2 ) factors into a product of the marginal cdfs, which in
turn by Theorem 2.4.2 implies thatX 1 andX 2 are independent.