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Continuous Probability Distributions

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.

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Continuous Probability Distributions

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  1. Continuous Probability Distributions A discrete random variable is a variable that can take on a countable number of possible values along a specified interval.

  2. Continuous Probability Distributions A continuous random variable is a variable that can take on any of the possible values between two points.

  3. 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

  4. Continuous Probability Distributions The probability distribution of a continuous random variable is represented by a probability density function that defines a curve.

  5. 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

  6. 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.

  7. Normal Probability Distribution NORMAL DISTRIBUTION DENSITY FUNCTION where: x = Any value of the continuous random variable such that - < x <  .  = Population standard deviation e = Base of the natural log = 2.7183  = Population mean

  8. Normal Probability Distribution(Figure 5-2) Probability = 0.50 Probability = 0.50 X  Mean Median Mode

  9. Difference Between Normal Distributions(Figure 5-3) x (a) x (b) x (c)

  10. 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.

  11. 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)

  12. Areas Under the Standard Normal Curve(Using Table 5-1) 0.1985 X 0 0.52 Example: z = 0.52 (or -0.52) A(z) = 0.1985 or 19.85%

  13. Areas Under the Standard Normal Curve(Table 5-1)

  14. 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

  15. 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

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