Using Simpson’s Rule for the normal distribution This problem uses Simpson’s rule to approximate a definite integral important in probability. In our probability unit, we learned that when given a probability density function f (x), we may compute the probability P that an event x is between a and b by calculating the definite integral: P (a ≤ x ≤ b) = b Z f (x) dx. a Here we’re assuming that a probability density function f (x) has the property that ∞ Z f (x) dx = 1. −∞ 2 √1 e−x π In the next session, we will show that f (x) = property. For now, we assume this property. is a probability density function with this Question: Suppose the probability density function for American male height is roughly (in inches x) h(x) = 1 √ 2.8 2π 2 /5.6 e−(x−69) . • Use Simpson’s rule to estimate the probability that an American male is between 5 and 6 feet tall. • Use Simpson’s rule to estimate the probability that an American male is over 8 feet tall. Solution: For the first part, 5 feet = 60 inches and 6 feet = 72 inches, so we must compute P (60 ≤ x ≤ 72) = Z 72 Z 72 h(x) dx = 60 60 1 √ 2.8 2π 2 /5.6 e−(x−69) dx 2 (NOTE: We’ve seen that e−x has no elementary antiderivative, so we can’t just compute the definite integral using the Fundamental Theorem of Calculus. Some numerical integration is required.) We can make a table of values of h(x) using a calculator (rounded to three decimal places): x 60 62 64 66 68 70 72 h(x) 7.45 ∗ 10−8 2.26 ∗ 10−5 1.60 ∗ 10−3 2.86 ∗ 10−2 0.119 0.119 2.86 ∗ 10−2 Now using Simpson’s rule, we estimate the definite integral to be: ∆x (h(60) + 4h(62) + 2h(64) + 4h(66) + 2h(68) + 4h(70) + h(72)) 3 1 where our width ∆x of each interval is 2 inches. This is approximately .574 (or 57.4 percent). For the second part, we need to make some assumptions about the normal distribution since, strictly speaking, the probability would be given by P (8 < x < ∞) = Z ∞ h(x) dx. 8 We don’t expect many people to be over 8 feet 4 inches. Indeed, h(100) is extremely small and h is decreasing. So we may estimate the above integral to high accuracy using the definite integral from x = 96 to x = 100, which in turn may be estimated by Simpson’s rule. Making a similar table: x h(x) 96 4.15 ∗ 10−58 98 8.55 ∗ 10−67 100 4.22 ∗ 10−76 Then Simpson’s rule estimates the integral: Z 100 h(x) dx = 96 2 (h(96) + 4h(98) + h(100)) = 2.77 ∗ 10−58 3 This is a very small number, and so even though there are over 300 million Americans, of which roughly half are male, we expect essentially no chance of seeing a person over 8 feet tall based on our model using the normal distribution. 2