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Continuous Probability Distributions Chapter Seven McGraw-Hill/Irwin © 2006 The McGraw-Hill Companies, Inc., All Rights Reserved. Continuous Probability Distributions The Random Variable (RV) that can take infinite values within a given range. •Probability is calculated for a range of values and not a specific value. •Eg. You don’t ask what is the probability drive time taken by students coming to CSUN is (exactly) 7 minutes? Rather, you ask, what is the probability drive time is between, say, 5 – 10 minutes? If the slots are infinitely narrow, then the chance of winning any one slot is zero (=1/∞) So, we will calculate probabilities for a range of values, rather than for a value of the RV. We will take the ratio of the area for the range of values to the total area. The Normal probability distribution s ∞ m ∞ Two characteristics determine a normal curve: Mean m Std. Deviation s Family of Normal Distributions See this in Visual Statistics Calculating area under a Normal curve Remember, in calculating probabilities for continuous RVs, we have to measure the area under the curve. Providing tables of areas for every possible normal curve is impractical (because too many combinations of m and s) . Fortunately, there is one member of the family that can be used to calculate areas for any normal curve. Its mean is 0 and standard deviation is 1. This curve is called the Standard Normal Distribution or the ZDistribution. (Appendix D, Page 496) Standard Normal Distribution or the Z-Distribution s=1 m=0 Question: How do we transform any given normal curve to a Standard Normal (or Z) distribution ? Slide Squish Remember the Transformation formula !!! X m z= s Notice that Z is a measure of how many SDs a given X value is from the mean. The original normal curve with mean m and s.d. s now becomes … … the Standard Normal Distribution with mean 0 and s.d. 1 ! Excel Example Source file: DemoX-MuBySigma.xls Problem on Page 197 Weekly income of a foreman is normally distributed with mean of $1000 and s.d. of $100. You select a foreman randomly? What is the probability that he/she earns between $1000-1100? Use the Z-transformation formula and compute the value of Z corresponding to X=1000 and X=1100. When X=1000, Z is (1000-1000)/100 = 0 When X=1100, Z is (1100-1000)/100 = 1 σ=100 σ=100 Slide μ=0 μ=1000 100 σ=1 Equivalent Std Normal Curve Z=0 1 1100 Proportion of area between dotted lines to the total area of the curve does not change with slide & squish operation! Go to Appendix D on Page 496 and find the area under the curve between Z=0 and Z=1. The area (which represents the probability) is 0.3413 Variation of the same problem ( Page 198) What is the probability of randomly selecting a foreman who earned less than $1100? The answer is: 0.8413 Another variation of the same problem (Page 198-199) a. What is the probability of selecting a foreman whose income is between $790-1000? Z = (790-1000)/100 = -2.10 From the Appendix table, this gives 0.4821 b. Less than $790? Z value is same as above. But we compute 0.5 – 0.4821 = 0.0179 What is the area between $840-1200? (page 200) Corresponding to $840, Z is (840-1000)/100 = -1.60 Corresponding to $1200, Z is (1200-1000)/100 = 2.0 Use Appendix D to find the total area. The answer is .4452+.4772 = 0.9224 Calculate the area under the curve between $11501250? Corresponding to $1250, Z is (1250-1000)/100 = 2.50 Corresponding to $1150, Z is (1150-1000)/100 = 1.50 Use the Appendix D to find the answer to be 0.0606