Continuous Random Variables: Continuous Probability Functions 1.5 2008/06/04 14:35:19 GMT-5 2008/07/31 12:19:17.682 GMT-5 Barbara Illowsky illowskybarbara@deanza.edu Susan Dean deansusan@deanza.edu Connexions cnx@cnx.org area continuous distribution elementary functions probability random statistics This module introduces the continuous probability function and explores the relationship between the probability of X and the area under the curve of f(X). We begin by defining a continuous probability density function. We use the function notation fX. Intermediate algebra may have been your first formal introduction to functions. In the study of probability, the functions we study are special. We define the function fX so that the area between it and the x-axis is equal to a probability. Since the maximum probability is one, the maximum area is also one.For continuous probability distributions, PROBABILITY = AREA.Consider the function fX=120 where 0 X 20 . X = a real number. The graph of fX=120 is a horizontal line. However, since 0 X 20 , fX is restricted to the portion between X0 and X20 . fX=120 where 0 X 20 . The graph of fX=120 is a horizontal line segment when 0 X 20 .The area between fX=120 where 0 X 20 and the x-axis is the area of a rectangle with base = 20 and height =120. AREA 20⋅120 1 This particular function, where we have restricted X so that the area between the function and the x-axis is 1, is an example of a continuous probability density function. It is used as a tool to calculate probabilities. Suppose we want to find the area between fX=120 and the x-axis where 0 X 2 . AREA (2-0)⋅ 120 0.1 (2-0) 2 base of a rectangle 120 = the height.The area corresponds to a probability. The probability that X is between 0 and 2 is 0.1, which can be written mathematically as P(0<X<2) = P(X<2)=0.1. Suppose we want to find the area between fX=120 and the x-axis where 4 X 15 . AREA (15-4)⋅ 120 0.55 (15-4) = 11 = the base of a rectangle120 = the height.The area corresponds to the probability P(4 X 15) 0.55. Suppose we want to find P(X = 15). On an x-y graph, X = 15 is a vertical line. A vertical line has no width (or 0 width). Therefore, P(X = 15) = (base)(height) = (0)(120) = 0. P(X x) (can be written as P(X x) for continuous distributions) is called the cumulative distribution function or CDF. Notice the "less than or equal to" symbol. We can use the CDF to calculate P(X x) . The CDF gives "area to the left" and P(X x) gives "area to the right." We calculate P(X x) for continuous distributions as follows: P(X x) 1-P(X x) . Label the graph with f(X) and X. Scale the x and y axes with the maximum x and y values. fX 120, 0 X 20 . P(2.3 X 12.7) (base) (height) (12.7-2.3) (120) 0.52The previous problem is an example of the uniform probability distribution.