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Continuous Probability Distributions

Continuous Probability Distributions. Many continuous probability distributions, including: Uniform Normal Gamma Exponential Chi-Squared Lognormal Weibull. Normal Distribution. The “ bell-shaped curve ” Also called the Gaussian distribution

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Continuous Probability Distributions

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  1. Continuous Probability Distributions • Many continuous probability distributions, including: • Uniform • Normal • Gamma • Exponential • Chi-Squared • Lognormal • Weibull ISE 327 Fall 2008

  2. Normal Distribution • The “bell-shaped curve” • Also called the Gaussian distribution • The most widely used distribution in statistical analysis • forms the basis for most of the parametric tests we’ll perform later in this course. • describes or approximates most phenomena in nature, industry, or research • Random variables (X) following this distribution are called normal random variables. • the parameters of the normal distribution are μand σ(sometimes μand σ2.) ISE 327 Fall 2008

  3. (μ = 5, σ = 1.5) Normal Distribution • The density function of the normal random variable X, with mean μ and variance σ2, is all x. ISE 327 Fall 2008

  4. Standard Normal RV … • Note: the probability of X taking on any value between x1 and x2 is given by: • To ease calculations, we define a normal random variable where Z is normally distributed with μ = 0 and σ2= 1 ISE 327 Fall 2008

  5. Standard Normal Distribution • Table A.1 : “Areas under the standard normal curve from - ∞ to z” • Page 915 negative values for z • Page 916 positive values for z ISE 327 Fall 2008

  6. Examples • P(Z ≤ 1) = • P(Z ≥ -1) = • P(-0.45 ≤ Z ≤ 0.36) = ISE 327 Fall 2008

  7. Your turn … • Use Table A.1 to determine (draw the picture!) 1. P(Z≤ 0.8) = 2. P(Z≥ 1.96) = 3. P(-0.25 ≤ Z≤ 0.15) = 4. P(Z ≤ -2.0 orZ≥ 2.0) = ISE 327 Fall 2008

  8. Applications of the Normal Distribution • A certain machine makes electrical resistors having a mean resistance of 40 ohms and a standard deviation of 2 ohms. What percentage of the resistors will have a resistance less than 44 ohms? • Solution: Xis normally distributed with μ = 40 and σ= 2 and x = 44 P(X<44) = P(Z< +2.0) = 0.9772 Therefore, we conclude that 97.72% will have a resistance less than 44 ohms. What percentage will have a resistance greater than 44 ohms? ISE 327 Fall 2008

  9. Terminology Used in ISE 327 Text • A certain machine makes electrical resistors having a mean resistance of 40 ohms and a standard deviation of 2 ohms. What percentage of the resistors will have a resistance greater than 44 ohms? • Solution: Xis normally distributed with μx= 40 and σx= 2 and x = 44 P(X>44) = 1 - P(Z< +2.0) = 1 - 0.9772 Therefore, we conclude that 2.28% will have a resistance greater than 44 ohms. ISE 327 Fall 2008

  10. Your Turn DRAW THE PICTURE!! • What is the probability that a single resistor will have a rating between 42 and 44 ohms? • Specifications are that the resistors are 40 ± 3 ohms. What percentage of the resistors will be within specifications? ISE 327 Fall 2008

  11. The Normal Distribution “In Reverse” • Example: Given a normal distribution with μ = 40 and σ = 6, find the value of X for which 45% of the area under the normal curve is to the left of X. • If P(Z < k) = 0.45, k = ___________ • Z = _______ X = _________ ISE 327 Fall 2008

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