3.10 Independence

Recall the definition of two independent events A and B:

(which is equivalent to P( A ? B) = P( A) P( B))

We extend this to independence of random variables as follows: X and Y are independent if

for every relevant value of X and Y, where g( x) is the marginal distribution of x, i.e., knowing that Y = y has no effect on the distribution of X.

From the definition of conditional probability it follows that

where h( y) is the marginal distribution of Y. In words: for discrete random variables the joint probability function is the product of the marginal probability functions. For continuous random variables the joint probability density function is the product of the marginal p.d.f.s.

Examples

1. Discrete random variables.

   Consider the joint distribution of fires and fire brigade calls given in Table 3.4. We have P( X = 1, Y = 1) = 0 .1. On the other hand, P( X = 1) = 0 .35 and P( Y = 1) = 0 .5. Thus P( X = 1, Y = 1) ? P( X = 1) P( Y = 1) and X and Y are not independent.

2. Continuous random variables.

   Consider example 2 of Section 3.9. In...