Continuous Probability Distributions Many continuous probability distributions including
Continuous Probability Distributions • Many continuous probability distributions, including: ü Uniform ü Normal ü Gamma ü Exponential ü Chi-Squared ü Lognormal ü Weibull Statistical Review for Chapters 3 and 4 ISE 327 Fall 2008 1
Normal Distribution • The “bell-shaped curve” • Also called the Gaussian distribution • The most widely used distribution in statistical analysis ü forms the basis for most of the parametric tests we’ll perform later in this course. ü describes or approximates most phenomena in nature, industry, or research • Random variables (X) following this distribution are called normal random variables. ü the parameters of the normal distribution are μ and σ (sometimes μ and σ2. ) Statistical Review for Chapters 3 and 4 ISE 327 Fall 2008 2
Normal Distribution • The density function of the normal random variable X, with mean μ and variance σ2, is all x. (μ = 5, σ = 1. 5) Statistical Review for Chapters 3 and 4 ISE 327 Fall 2008 3
Standard Normal RV … • Note: the probability of X taking on any value between x 1 and x 2 is given by: • To ease calculations, we define a normal random variable where Z is normally distributed with μ = 0 and σ2 = 1 Statistical Review for Chapters 3 and 4 ISE 327 Fall 2008 4
Standard Normal Distribution • Table A. 1 : “Areas under the standard normal curve from - ∞ to z” ü Page 915 negative values for z ü Page 916 positive values for z Statistical Review for Chapters 3 and 4 ISE 327 Fall 2008 5
Examples • P(Z ≤ 1) = • P(Z ≥ -1) = • P(-0. 45 ≤ Z ≤ 0. 36) = Statistical Review for Chapters 3 and 4 ISE 327 Fall 2008 6
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 or Z ≥ 2. 0) = Statistical Review for Chapters 3 and 4 ISE 327 Fall 2008 7
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: X is 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? Statistical Review for Chapters 3 and 4 ISE 327 Fall 2008 8
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: X is normally distributed with μ x= 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. Statistical Review for Chapters 3 and 4 ISE 327 Fall 2008 9
Your Turn DRAW THE PICTURE!! 1. What is the probability that a single resistor will have a rating between 42 and 44 ohms? 2. Specifications are that the resistors are 40 ± 3 ohms. What percentage of the resistors will be within specifications? Statistical Review for Chapters 3 and 4 ISE 327 Fall 2008 10
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. 1) If P(Z < k) = 0. 45, k = ______ 2) Z = _______ X = _____ Statistical Review for Chapters 3 and 4 ISE 327 Fall 2008 11
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