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 JMB Chapter 6 Lecture 2 EGR 252 Spring 2010 Slide 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. ) JMB Chapter 6 Lecture 2 EGR 252 Spring 2010 Slide 2
Normal Distribution The density function of the normal random variable X, with mean μ and variance σ2, has the following properties: § peak is both the mean and the mode and occurs at x = μ § curve is symmetrical about a vertical axis through the mean § total area under the curve and above the horizontal axis = 1. § points of inflection are at x = μ + σ (μ = 5, σ = 1. 5) JMB Chapter 6 Lecture 2 EGR 252 Spring 2010 Slide 3
Standard Normal Random Variable • 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 JMB Chapter 6 Lecture 2 EGR 252 Spring 2010 Slide 4
Standard Normal Distribution • Table A. 3: “Areas Under the Normal Curve” JMB Chapter 6 Lecture 2 EGR 252 Spring 2010 Slide 5
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? JMB Chapter 6 Lecture 2 EGR 252 Spring 2010 Slide 6
The Normal Distribution “In Reverse” • Example: Given a normal distribution with = 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. • Solution: If P(Z < k) = 0. 45, what is the value of k? 45% of area under curve less than k k = -0. 125 from table in back of book Z values Z = (x- μ) / σ Z = -0. 125 = (x-40) / 6 X = 39. 25 40 X values JMB Chapter 6 Lecture 2 EGR 252 Spring 2010 Slide 7
Unscheduled Quiz • P(Z ≤ 1) = Area to the left of the blue line • P(Z ≥ -1) = Area to the right of the blue line • P(-2. 5 ≤ Z ≤ 2. 8) = Area between the two lines • For a normally distributed variable for which the mean is 30 and standard deviation is 10, P(X < 40) = Area to the left of the blue line JMB Chapter 6 Lecture 2 30 40 x values for µ = 30 EGR 252 Spring 2010 Slide 8
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