CONTINUOUS RANDOM VARIABLES

In contrast to a discrete random variable, a continuous random variable can take on any value within a given range. A good example of a continuous random variable is the return of a stock index. If the level of the index can be any real number between zero and infinity, then the return of the index can be any real number greater than -1.

Even if the range that the continuous variable occupies is finite, the number of values that it can take is infinite. For this reason, for a continuous variable, the probability of any specific value occurring is zero.

Even though we cannot talk about the probability of a specific value occurring, we can talk about the probability of a variable being within a certain range. Take, for example, the return on a stock market index over the next year. We can talk about the probability of the index return being between 6% and 7%, but talking about the probability of the return being exactly 6.001% or exactly 6.002% is meaningless. Even between 6.001% and 6.002% there are literally an infinite number of possible values. The probability of any one of those infinite values occurring is zero.