Probability, Random Variables, and Random Processes: Theory and Signal Processing Applications

4.4 INDEPENDENT RANDOM VARIABLES

Independent random variables arise in many applications, and it is often assumed as an approximation in order to derive results that otherwise would not be possible in some problems.

Definition: Independent Random Variables Random variables X and Y are independent if and only if

(4.27)

where fX(x) and FY(y) are the marginal cdfs.

Since a cdf is the probability of an event, this definition is consistent not only with intuition but also from the definition based on probabilities in Chapter 2: P(AB) = P(A)P(B) for independent events A and B. It is straightforward to show from (4.27) that the joint pdf also splits into a product:

(4.28)

for independent random variables X and Y (we could also state this as the definition of independence).

Example 4.3. Let X be the standard Gaussian random variable and Y be exponential with parameter λ. If they are independent, then we can immediately write their joint (bivariate) pdf as

(4.29)

This can be extended to additional independent random variables. If Z is uniform on [−1, 1], then the joint pdf of all three random variables is

(4.30)

which is the previous expression scaled by 1/2, and it includes an indicator ...