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Chapter 5. Continuous Probability Distributions. Chapter 5 - Chapter Outcomes. After studying the material in this chapter, you should be able to: • Discuss the important properties of the normal probability distribution.

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## Chapter 5

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**Chapter 5**Continuous Probability Distributions**Chapter 5 - Chapter Outcomes**After studying the material in this chapter, you should be able to: • Discuss the important properties of the normal probability distribution. • Recognize when the normal distribution might apply in a decision-making process.**Chapter 5 - Chapter Outcomes(continued)**After studying the material in this chapter, you should be able to: • Calculate probabilities using the normal distribution table and be able to apply the normal distribution in appropriate business situations. • Recognize situations in which the uniform and exponential distributions apply.**Continuous Probability Distributions**A discrete random variable is a variable that can take on a countable number of possible values along a specified interval.**Continuous Probability Distributions**A continuous random variable is a variable that can take on any of the possible values between two points.**Examples of Continuous Random variables**• Time required to perform a job • Financial ratios • Product weights • Volume of soft drink in a 12-ounce can • Interest rates • Income levels • Distance between two points**Continuous Probability Distributions**The probability distribution of a continuous random variable is represented by a probability density function that defines a curve.**Continuous Probability Distributions**(a) Discrete Probability Distribution (b) Probability Density Function P(x) f(x) x x Possible Values of x Possible Values of x**Normal Probability Distribution**The Normal Distribution is a bell-shaped, continuous distribution with the following properties: 1. It is unimodal. 2. It is symmetrical; this means 50% of the area under the curve lies left of the center and 50% lies right of center. 3. The mean, median, and mode are equal. 4. It is asymptotic to the x-axis. 5. The amount of variation in the random variable determines the width of the normal distribution.**Normal Probability Distribution**NORMAL DISTRIBUTION DENSITY FUNCTION where: x = Any value of the continuous random variable = Population standard deviation e = Base of the natural log = 2.7183 = Population mean**Normal Probability Distribution(Figure 5-2)**Probability = 0.50 Probability = 0.50 f(x) x Mean Median Mode**Differences Between Normal Distributions(Figure 5-3)**x (a) x (b) x (c)**Standard Normal Distribution**The standard normal distribution is a normal distribution which has a mean = 0.0 and a standard deviation = 1.0. The horizontal axis is scaled in standardized z-values that measure the number of standard deviations a point is from the mean. Values above the mean have positive z-values and those below have negative z-values.**Standard Normal Distribution**STANDARDIZED NORMAL Z-VALUE where: x = Any point on the horizontal axis = Standard deviation of the normal distribution = Population mean z = Scaled value (the number of standard deviations a point x is from the mean)**Areas Under the Standard Normal Curve(Using Table 5-1)**0.1985 X 0 0.52 Example: z = 0.52 (or -0.52) P(0 < z < .52) = 0.1985 or 19.85%**Standard Normal Example(Figure 5-6)**Probabilities from the Normal Curve for Westex 0.1915 0.50 x z x=45 50 z= -.50 0**Standard Normal Example(Figure 5-7)**z z=-1.25 x=7.5 From the normal table: P(-1.25 z 0) = 0.3944 Then, P(x 7.5 hours) = 0.50 - 0.3944 = 0.1056**Uniform Probability Distribution**The uniform distribution is a probability distribution in which the probability of a value occurring between two points, a and b, is the same as the probability between any other two points, c and d, given that the distribution between a and b is equal to the distance between c and d.**Uniform Probability Distribution**CONTINUOUS UNIFORM DISTRIBUTION where: f(x) = Value of the density function at any x value a = Lower limit of the interval from a to b b = Upper limit of the interval from a to b**Uniform Probability Distributions(Figure 5-16)**f(x) f(x) for 2 x 5 for 3 x 8 .50 .50 .25 .25 2 5 3 8 a b a b**Exponential Probability Distribution**The exponential probability distribution is a continuous distribution that is used to measure the time that elapses between two occurrences of an event.**Exponential Probability Distribution**EXPONENTIAL DISTRIBUTION A continuous random variable that is exponentially distributed has the probability density function given by: where: e = 2.71828. . . 1/ = The mean time between events ( >0)**Exponential Distributions(Figure 5-18)**Lambda = 3.0 (Mean = 0.333) f(x) Lambda = 2.0 (Mean = 0.5) Lambda = 1.0 (Mean = 1.0) Lambda = 0.50 (Mean = 020) x Values of x**Exponential Probability**EXPONENTIAL PROBABILITY**• Continuous Random Variable**• Discrete Random Variable • Exponential Distribution • Normal Distribution • Standard Normal Distribution Standard Normal Table • Uniform Distribution • z-Value Key Terms

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