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 PDF JMB Ch 6 Excel and Z table Source: www. itl. nist. gov EGR 252 2018 Slide 1
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 JMB Ch 6 Excel and Z table EGR 252 2018 Slide 2
Standard Normal Distribution • Table A. 3 Pages 735 -736: “Areas under the Normal Curve” JMB Ch 6 Excel and Z table EGR 252 2018 Slide 3
JMB Ch 6 Excel and Z table EGR 252 2018 Slide 4
Examples • P(Z ≤ 1) = • P(Z ≥ -1) = • P(-0. 45 ≤ Z ≤ 0. 36) = JMB Ch 6 Excel and Z table EGR 252 2018 Slide 5
Name: ____________ • Use Table A. 3 to determine (draw the picture!) 1. P(Z ≤ 0. 8) = 0. 788145=NORM. S. DIST(0. 8, TRUE) 2. P(Z ≥ 1. 96) = 0. 024998=1 -NORM. S. DIST(1. 96, TRUE) JMB Ch 6 Excel and Z table EGR 252 2018 Slide 6
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 Ch 6 Excel and Z table EGR 252 2018 Slide 7
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. Step 1 If P(Z < z) = 0. 45, z = _______ (from Table A. 3) Why is z negative? Step 2 45% X = _____ 39. 24603=NORM. INV(0. 45, 40, 6) JMB Ch 6 Excel and Z table EGR 252 2018 Slide 8
- Slides: 8