A discrete random variable is a variable that has no single value. Continuous discrete random variables are bound by events. This continuity idea on the variable is important, as it indicates that we can not calculate the probability of a single event, in fact we can only calculate the probability that the event will occur between a the bounds of the variable. For example, X is a continuous random variable, and its probability P, is: P(1 <= X <= 3). It is not possible to find the probability of X=2, but it is possible to find the probability of the events between one and three.
The basic case of calculus applied to a probability is a uniform density function. A uniform density function is a function representing that all event have the same chance to occur. This function is represented in the form " Y = k" where k is a numeric constant.
The graph looks like a horizontal line parallel to the "X" axis, and bound by the events it represents.
A probability density function is a function whose area above the "X" axis is equal to one, and which represents events and their individual chance to occur. Since every event has a different chance to occur, the function is a curve. A normal probability density function has a bell shape, and extends from negative infinite to positive infinite. Other density functions extend from zero to positive infinity, such as the chi square distribution. Portions of the area under the function are used to represent the probability of events.
Probability density functions represent events and their probability to occur as an area under the function. The area of the complete function is equal to 1 -- or in percentages, 100 percent. Therefore, portions of the area will numbers be between zero and one -- or zero and 100 percent. Integrals are used to find the area under a curve; therefore, by setting up an integral bounded by events, the integral can calculate the area, returning the probability of such events occurring.