## Correlation

If two random variables bear some relationship to one another, they may be correlated. This notion has a specific meaning in probability theory. First, we define a term known as the covariance of two distributions, given as

Cov(xy) = E{[x - E(x)][y - E(y)]} = E(xy) - E(x)E(y) (11.20)

Clearly, if the two variables are independent, then E(xy) = E(x)E(y) and the covariance reduces to zero.

For correlated random variables, we may also inquire into the effect on the variance of the sum of such variables, that is, var(x + y). In such a case, the variance has a covariance term, as var(x + y) = var(x) + var(y) + 2 Cov(xy) (11.21)

A formal correlation coefficient may also be calculated from the definition:

By dividing by the product of the individual standard deviations, the correlation coefficient is normalized to values between - 1 and + 1.

### 11.3 THE BINOMIAL DISTRIBUTION

The specific discrete distribution known as the binomial distribution arises when there are repeated independent trials with only two resultant possibilities. If we call the probability of success p and the probability of failure q, where (p + q = 1), then the distribution becomes

= 0, otherwise

This distribution defines the probability of exactly x successes in n independent trials.

Example. We often assume that the bits of data that are transmitted in a digital communication system are independent from one another and that there is a bit error rate (BER) of some value, for example, 10"8. For purposes of illustrating the binomial, we assume a character that is 8 bits long and that the BER is 10~3. If receiving an individual bit without error is defined as success, then the probability of ''success'' is 0.999 and the probability of ''failure'' is 0.001. The probability that we will have eight successes (no errors in the transmission of a character) is then

In a similar fashion, one can then calculate the probabilities of no error in a set of bits (a byte), characters (a word), a series of words (a message), and so forth. Example. If, when throwing a die, an odd number is success and an even number failure, the probability of exactly four successes in ten trials is

P(4) = (10) (0.5)4(0.5)6 = (210)(0.0625)(0.015625) = 0.205

The mean value of the binomial is equal to np, which is the expected number of successes in n trials. This is in consonance with our intuition as we, for example, would expect to have 40 successes in 100 trials if the probability of success on each trial were 0.4. 