Chapter 6

Continuous Probability Distributions

Figure 6.1 A Discrete Probability Distribution Probability is represented by the height of the bar.Holes or breaks between values 1 2 3 4 5 6 7 xP(x)

Figure 6.2 A Continuous Probability Distribution

Figure 6.3 Marias Commute Time Distribution

Figure 6.4 Computing P(64 < x < 67) Probability = Area = Width x Height = 3 x 1/20 = .15 or 15%

Figure 6.5 Total Area = 1.0 1/20

Uniform Probability Density Function (6.1) f(x) =1/(b-a) for a < x < b0 everywhere else

Figure 6.7 The Bell-Shaped Normal Distribution Mean is mStandard Deviation is sx

Normal Probability Density Function (6.5) f(x) =

Normal Distribution PropertiesApproximately 68.3% of the values in a normal distribution will be within one standard deviation of the distribution mean, m.

Approximately 95.5% of the values in a normal distribution will be within two standard deviations of the distribution mean, m.

Approximately 99.7% of the values (nearly all of them) in a normal distribution will be found within three standard deviations of the distribution mean, m.

Figure 6.8 Normal Area for m + 1s

Figure 6.9 Normal Area for m + 2s m-2s m m+2s

Figure 6.10 Normal Area for m + 3s

Standard Normal Table

Normal Table (2)

Normal Table (3)

Normal Table (4)

Figure 6.11 Area in a Standard Normal Distribution for a z of +1.2 0 1.2 z.3849

z-score Calculation (6.6) z =

Figure 6.12 P(50 < x < 55)

Figure 6.13 P(x > 58) 50 58 x (diameter) .4772 0 2.0 zm =50s = 4.0228

Figure 6.14 P(47 < x < 56)

Figure 6.15 P(54 < x < 58)

Figure 6.16 Finding the z score for an Area of .25 0 ? z.2500

Exponential Probability (6.7) Density Function f(x) =

Figure 6.17 Exponential Probability Density Function f(x) =l/elxf(x)x

Calculating Area for the (6.8) Exponential Distribution P(x > a) =

Figure 6.18 Calculating Areas for the Exponential Distribution

Figure 6.19 Finding P(1.0 < x < 1.5)

Figure 6.20 Finding P(1.0 < x < 1.5)

Expected Value for the (6.9) Exponential Distribution E(x) =

Variance for the Exponential (6.10) Distribution 2s2 =

Standard Deviation for the (6.11) Exponential Distribution s = =