The Density Function Help (page 2)

By — McGraw-Hill Professional
Updated on Aug 26, 2011

Area Under the Curve

Figure 3-8 is an expanded view of the curve derived by ''refining the points'' of Fig. 3-7 to their limit. This density function, like all density functions, has a special property: if you calculate or measure the total area under the curve, it is always equal to 1. This rule holds true for the same reason that the relative frequency values of the outcomes for a discrete variable always add up to 1 (or 100%), as we learned in the last chapter.

The Density Function

Consider two hypothetical systolic blood pressure values: say a and b as shown in Fig. 3-8. (Again, we refrain from giving specific numbers because this example is meant to be for illustration, not to represent any recorded fact.) The probability that a randomly chosen person will have a systolic blood pressure reading k that is between a and b can be written in any of four ways:

      P(a < k < b)
      P(ak < b)
      P(a < kb)

The first of these expressions includes neither a nor b, the second includes a but not b, the third includes b but not a, and the fourth includes both a and b. All four expressions are identical in the sense that they are all represented by the shaded portion of the area under the curve. Because of this, an expression with less-than signs only is generally used when expressing discrete probability.

If the vertical lines x = a and x = b are moved around, the area of the shaded region gets larger or smaller. This area can never be less than 0 (when the two lines coincide) or greater than 1 (when the two lines are so far apart that they allow for the entire area under the curve).

Practice problems for these concepts can be found at:

Basics of Probability Practice Test

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