Independent Random Variables
When a study involves pairs of
random variables,
it is often useful to know
whether or not the random variables are independent. This lesson
explains how to assess the independence of random variables.
Independence of Random Variables
If two random variables, X and Y, are independent,
they satisfy the following conditions.
 P(xy) = P(x), for all values of X and Y.
 P(x ∩ y)
= P(x) * P(y), for all values of X and Y.
The above conditions are equivalent. If either one is met, the
other condition also met; and X and Y are
independent. If either condition is not met,
X and Y are dependent.
Note: If X and Y are independent, then the
correlation
between X and Y is equal to zero.
Joint Probability Distributions

X

0 
1 
2 
Y 
3 
0.1 
0.2 
0.2 
4 
0.1 
0.2 
0.2 
The table on the right shows the joint probability distribution between
two discrete random variables  X and Y.
In a joint probability distribution
table, numbers in the cells of the table represent the probability
that particular values of X and Y occur together. From this table,
you can see that the probability that X=0 and Y=3 is 0.1; the probability
that X=1 and Y=3 is 0.2; and so on.
You can use tables like this to figure out whether two discrete
random variables are independent or dependent. Problem 1 below
shows how.
Test Your Understanding
Problem 1
The table on the left shows the joint probability distribution between
two random variables  X and Y; and the table on the right shows
the joint probability distribution between two random variables  A
and B.

X

0 
1 
2 
Y 
3 
0.1 
0.2 
0.2 
4 
0.1 
0.2 
0.2 


A

0 
1 
2 
B 
3 
0.1 
0.2 
0.2 
4 
0.2 
0.2 
0.1 

Which of the following statements are true?
I. X and Y are independent random variables.
II. A and B are independent random variables.
(A) I only
(B) II only
(C) I and II
(D) Neither statement is true.
(E) It is not possible to answer this question, based on the
information given.
Solution
The correct answer is A. The solution requires several computations
to test the independence of random variables. Those computations are
shown below.
X and Y are independent if P(xy) = P(x), for all values of X and Y.
From the probability distribution table, we know the following:
P(x=0) = 0.2; P(x=0  y=3) = 0.2; P(x=0  y = 4) = 0.2
P(x=1) = 0.4; P(x=1  y=3) = 0.4; P(x=1  y = 4) = 0.4
P(x=2) = 0.4; P(x=2  y=3) = 0.4; P(x=2  y = 4) = 0.4
Thus, P(xy) = P(x), for all values of X and Y, which means that
X and Y are independent. We repeat the same analysis to test
the independence of A and B.
P(a=0) = 0.3; P(a=0  b=3) = 0.2; P(a=0  b = 4) = 0.4
P(a=1) = 0.4; P(a=1  b=3) = 0.4; P(a=1  b = 4) = 0.4
P(a=2) = 0.3; P(a=2  b=3) = 0.4; P(a=2  b = 4) = 0.2
Thus, P(ab) is not equal to P(a), for all values of A and B. For
example, P(a=0) = 0.3; but P(a=0  b=3) = 0.2. This
means that A and B are not independent.