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8.6: Continuous Probability Functions

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    We begin by defining a continuous probability density function. We use the function notation \(f(x)\). 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 \(f(x)\) 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.

    Example \(\PageIndex{1}\)

    Consider the function \(f(x) = \frac{1}{20}\) for \(0 \leq x \leq 20\). \(x =\) a real number. The graph of \(f(x) = \frac{1}{20}\) is a horizontal line. However, since \(0 \leq x \leq 20\), \(f(x)\) is restricted to the portion between \(x = 0\) and \(x = 20\), inclusive.

    This shows the graph of the function f(x) = 1/20. A horiztonal line ranges from the point (0, 1/20) to the point (20, 1/20). A vertical line extends from the x-axis to the end of the line at point (20, 1/20) creating a rectangle. Figure \(\PageIndex{1}\)

    \[f(x) = \frac{1}{20} \text{ for } 0 \leq x \leq 20.\]

    The graph of \(f(x) = \frac{1}{20}\) is a horizontal line segment when \(0 \leq x \leq 20\).

    The area between \(f(x) = \frac{1}{20}\) where \(0 \leq x \leq 20\) and the x-axis is the area of a rectangle with base = 20 and height = \(\frac{1}{20}\).

    \[AREA = 20 \left(\frac{1}{20} \right) = 1\]

    Suppose we want to find the area between \(f(x) = \frac{1}{20}\) and the x-axis where \(0 < x < 2\).

    5.1.2.pngThis shows the graph of the function f(x) = 1/20. A horiztonal line ranges from the point (0, 1/20) to the point (20, 1/20). A vertical line extends from the x-axis to the end of the line at point (20, 1/20) creating a rectangle. A region is shaded inside the rectangle from x = 0 to x = 2. Figure \(\PageIndex{2}\)

    \[AREA = (2 - 0) \left(\dfrac{1}{20} \right) = 0.1\]

    \((2 - 0) = 2 = \text{base of a rectangle}\)

    REMINDER: area of a rectangle = (base)(height).

    The area corresponds to a probability. The probability that x is between zero and two 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 \(f(x) = \frac{1}{20}\) and the x-axis where \(4 < x < 15\).

    This shows the graph of the function f(x) = 1/20. A horiztonal line ranges from the point (0, 1/20) to the point (20, 1/20). A vertical line extends from the x-axis to the end of the line at point (20, 1/20) creating a rectangle. A region is shaded inside the rectangle from x = 4 to x = 15. Figure \(\PageIndex{3}\)

    \(\text{AREA} = (15 – 4)(\frac{1}{20}) = 0.55\)

    \(\text{AREA} = (15 – 4)(\frac{1}{20}) = 0.55\)

    \((15 – 4) = 11 = \text{the base of a rectangle}(15 – 4) = 11 = \text{the base of a rectangle}\)

    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 zero width). Therefore, \(P(x = 15) = (\text{base})(\text{height}) = (0)\left(\frac{1}{20}\right) = 0\)

    This shows the graph of the function f(x) = 1/20. A horiztonal line ranges from the point (0, 1/20) to the point (20, 1/20). A vertical line extends from the x-axis to the end of the line at point (20, 1/20) creating a rectangle. A vertical line extends from the horizontal axis to the graph at x = 15. Figure \(\PageIndex{4}\)

    \(P(X \leq x)\) (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: \(P(X > x) = 1 – P(X < x)\).

    This shows the graph of the function f(x) = 1/20. A horiztonal line ranges from the point (0, 1/20) to the point (20, 1/20). A vertical line extends from the x-axis to the end of the line at point (20, 1/20) creating a rectangle. The area to the left of a value, x, is shaded. Figure \(\PageIndex{5}\)

    Label the graph with \(f(x)\) and \(x\). Scale the \(x\) and \(y\) axes with the maximum \(x\) and \(y\) values. \(f(x) = \frac{1}{20}\), \(0 \leq x \leq 20\).

    To calculate the probability that \(x\) is between two values, look at the following graph. Shade the region between \(x = 2.3\) and \(x = 12.7\). Then calculate the shaded area of a rectangle.

    This shows the graph of the function f(x) = 1/20. A horiztonal line ranges from the point (0, 1/20) to the point (20, 1/20). A vertical line extends from the x-axis to the end of the line at point (20, 1/20) creating a rectangle. A region is shaded inside the rectangle from x = 2.3 to x = 12.7 Figure \(\PageIndex{6}\)

    \[P(2.3 < x < 12.7) = (\text{base})(\text{height}) = (12.7−2.3)\left(\dfrac{1}{20}\right) = 0.52\]

    Exercise \(\PageIndex{1}\)

    Consider the function \(f(x) = \frac{1}{8}\) for \(0 \leq x \leq 8\). Draw the graph of \(f(x)\) and find \(P(2.5 < x < 7.5)\).

    Answer
    5.1.7.png Figure \(\PageIndex{7}\)

    \(P (2.5 < x < 7.5) = 0.625\)

    Summary

    The probability density function (pdf) is used to describe probabilities for continuous random variables. The area under the density curve between two points corresponds to the probability that the variable falls between those two values. In other words, the area under the density curve between points a and b is equal to \(P(a < x < b)\). The cumulative distribution function (cdf) gives the probability as an area. If \(X\) is a continuous random variable, the probability density function (pdf), \(f(x)\), is used to draw the graph of the probability distribution. The total area under the graph of \(f(x)\) is one. The area under the graph of \(f(x)\) and between values a and b gives the probability \(P(a < x < b)\).

    The graph on the left shows a general density curve, y = f(x). The region under the curve and above the x-axis is shaded. The area of the shaded region is equal to 1. This shows that all possible outcomes are represented by the curve. The graph on the right shows the same density curve. Vertical lines x = a and x = b extend from the axis to the curve, and the area between the lines is shaded. The area of the shaded region represents the probabilit ythat a value x falls between a and b. Figure \(\PageIndex{8}\)

    The cumulative distribution function (cdf) of \(X\) is defined by \(P(X \leq x)\). It is a function of \(x\) that gives the probability that the random variable is less than or equal to \(x\).

    Formula Review

    Probability density function (pdf) \(f(x)\):

    • \(f(x) \geq 0\)
    • The total area under the curve \(f(x)\) is one.

    Cumulative distribution function (cdf): \(P(X \leq x)\)

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