Continuous Random Variables

As the number of values of X increases in a given range of values, the spacing between them becomes smaller, so small in the limit that one cannot distinguish between one unique value and another. So also do the value's individual probabilities become arbitrarily small in order not to violate the rule about all probabilities adding up to 1. Such a random variable is called a continuous random variable because there is literally no space between one value and another; one value flows continuously to the next. Curiously, the probability of a specific value is arbitrarily near but not equal to 0. However, over a small range, say from X₁ to X₁ + dX, the probability of X being in this range is not necessarily small. ^[8]

As the number of elements in the probability function becomes arbitrarily large, the ∑ morphs smoothly to the integral ∫: ∫_a-b all f(X) dX means integrate over all continuous values of X from values of a_lower to b_upper

∫_a-b all f(X) dX = 1

There are any number of continuous random variables in projects, or random variables that are so nearly continuous as to be reasonably thought of as continuous. The actual cost range of a work breakdown structure work package, discrete perhaps to the penny but for most practical applications continuous, is one example. Schedule duration range is another if measured to arbitrarily small units of time. Lifetime ranges of tools, facilities, and components are generally thought of as continuous.