1 Special Continuous Probability Distributions -Normal Distributions -Lognormal Distributions Dr. Jerrell T. Stracener, SAE Fellow Leadership in Engineering EMIS 7370/5370 STAT 5340 : PROBABILITY AND STATISTICS FOR SCIENTISTS AND ENGINEERS Systems Engineering Program Department of Engineering Management, Information and Systems Stracener_EMIS 7370/STAT 5340_Sum 08_

2 A random variable X is said to have a normal (or Gaussian) distribution with parameters  and , where -  0, with probability density function for -  < x <  f(x) x Normal Distribution

Stracener_EMIS 7370/STAT 5340_Sum 08_ the effects of  and  Properties of the Normal Model

Stracener_EMIS 7370/STAT 5340_Sum 08_ Mean or expected value of Mean = E(X) =  Median value of X 0.5 =  Standard deviation Normal Distribution

Stracener_EMIS 7370/STAT 5340_Sum 08_ Standard Normal Distribution If ~ N( ,  ) and if then Z ~ N(0, 1). A normal distribution with  = 0 and  = 1, is called the standard normal distribution. Normal Distribution

Stracener_EMIS 7370/STAT 5340_Sum 08_  x 0 z f(x) f(z) P (X<x ’ ) = P (Z<z ’ ) X’ Z’ Normal Distribution

Stracener_EMIS 7370/STAT 5340_Sum 08_ Standard Normal Distribution Table of Probabilities Enter table with and find the value of  Excel z 0  zz f(z) Normal Distribution

Stracener_EMIS 7370/STAT 5340_Sum 08_ The following example illustrates every possible case of application of the normal distribution. Let ~ N(100, 10) Find: (a) P(X < 105.3) (b) P(X  91.7) (c) P(87.1 <  115.7) (d) the value of x for which P(  x) = 0.05 Normal Distribution - Example

Stracener_EMIS 7370/STAT 5340_Sum 08_ a. P( < 105.3) = = P( < 0.53) = F(0.53) = x 0 z f(x) f(z) Normal Distribution – Example Solution

Stracener_EMIS 7370/STAT 5340_Sum 08_ b. P(  91.7) = = P( -0.83) = 1 - P( < -0.83) = 1- F(-0.83) = = x 0 z f(x) f(z) Normal Distribution – Example Solution

Stracener_EMIS 7370/STAT 5340_Sum 08_ c. P(87.1 <  115.7)= F(115.7) - F(87.1) = P(-1.29 < Z < 1.57) = F(1.57) - F(-1.29) = = x f(x) x f(x) Normal Distribution – Example Solution

Stracener_EMIS 7370/STAT 5340_Sum 08_ x 0 z f(x) f(z) Normal Distribution – Example Solution

Stracener_EMIS 7370/STAT 5340_Sum 08_ (d) P(  x) = 0.05 P(  z) = 0.05implies that z = 1.64 P(  x) = therefore x = 16.4 x = Normal Distribution – Example Solution

Stracener_EMIS 7370/STAT 5340_Sum 08_ The time it takes a driver to react to the brake lights on a decelerating vehicle is critical in helping to avoid rear-end collisions. The article ‘Fast-Rise Brake Lamp as a Collision-Prevention Device’ suggests that reaction time for an in-traffic response to a brake signal from standard brake lights can be modeled with a normal distribution having mean value 1.25 sec and standard deviation 0.46 sec. What is the probability that reaction time is between 1.00 and 1.75 seconds? If we view 2 seconds as a critically long reaction time, what is the probability that actual reaction time will exceed this value? Normal Distribution – Example Solution

Stracener_EMIS 7370/STAT 5340_Sum 08_ Normal Distribution – Example Solution

Stracener_EMIS 7370/STAT 5340_Sum 08_ Normal Distribution – Example Solution

Stracener_EMIS 7370/STAT 5340_Sum 08_ Statistical Quality Control is an application of probabilitistic and statistical techniques to quality control An Application of Probability & Statistics Statistical Quality Control

Stracener_EMIS 7370/STAT 5340_Sum 08_ Statistical Quality Control - Elements Analysis of process capability Statistical process control Process improvement Acceptance sampling

Stracener_EMIS 7370/STAT 5340_Sum 08_ Statistical Tolerancing - Convention Normal Probability Distribution LTL Nominal UTL +3 -3 

Stracener_EMIS 7370/STAT 5340_Sum 08_ Statistical Tolerancing - Concept LTL UTLNominal x

Stracener_EMIS 7370/STAT 5340_Sum 08_ For a normal distribution, the natural tolerance limits include 99.73% of the variable, or put another way, only 0.27% of the process output will fall outside the natural tolerance limits. Two points should be remembered: % outside the natural tolerances sounds small, but this corresponds to 2700 nonconforming parts per million. 2. If the distribution of process output is non normal, then the percentage of output falling outside   3  may differ considerably from 0.27%. Caution

Stracener_EMIS 7370/STAT 5340_Sum 08_ The diameter of a metal shaft used in a disk-drive unit is normally distributed with mean inches and standard deviation inches. The specifications on the shaft have been established as  inches. We wish to determine what fraction of the shafts produced conform to specifications. Normal Distribution - Example

Stracener_EMIS 7370/STAT 5340_Sum 08_ USL LSL f(x) x nominal Normal Distribution - Example Solution

Stracener_EMIS 7370/STAT 5340_Sum 08_ Thus, we would expect the process yield to be approximately 91.92%; that is, about 91.92% of the shafts produced conform to specifications. Note that almost all of the nonconforming shafts are too large, because the process mean is located very near to the upper specification limit. Suppose we can recenter the manufacturing process, perhaps by adjusting the machine, so that the process mean is exactly equal to the nominal value of Then we have Normal Distribution - Example Solution

Stracener_EMIS 7370/STAT 5340_Sum 08_ USL LSL f(x) x nominal Normal Distribution - Example Solution

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Stracener_EMIS 7370/STAT 5340_Sum 08_ Using a normal probability distribution as a model for a quality characteristic with the specification limits at three standard deviations on either side of the mean. Now it turns out that in this situation the probability of producing a product within these specifications is , which corresponds to 2700 parts per million (ppm) defective. This is referred to as three-sigma quality performance, and it actually sounds pretty good. However, suppose we have a product that consists of an assembly of 100 components or parts and all 100 parts must be non-defective for the product to function satisfactorily. Normal Distribution - Example

Stracener_EMIS 7370/STAT 5340_Sum 08_ The probability that any specific unit of product is non-defective is x x... x = (0.9973) 100 = That is, about 23.7% of the products produced under three sigma quality will be defective. This is not an acceptable situation, because many high technology products are made up of thousands of components. An automobile has about 200,000 components and an airplane has several million! Normal Distribution - Example

Stracener_EMIS 7370/STAT 5340_Sum 08_ Lognormal Distribution

Stracener_EMIS 7370/STAT 5340_Sum 08_ Definition - A random variable is said to have the Lognormal Distribution with parameters  and , where  > 0 and  > 0, if the probability density function of X is:,for x > 0,for x  0 x f(x) 0 Lognormal Distribution

Stracener_EMIS 7370/STAT 5340_Sum 08_ Rule:If ~ LN( ,  ), then = ln ( ) ~ N( ,  ) Probability Distribution Function where F(z) is the cumulative probability distribution function of N(0,1) Lognormal Distribution - Properties

Stracener_EMIS 7370/STAT 5340_Sum 08_ Mean or Expected Value Standard Deviation Median Lognormal Distribution - Properties

Stracener_EMIS 7370/STAT 5340_Sum 08_ A theoretical justification based on a certain material failure mechanism underlies the assumption that ductile strength X of a material has a lognormal distribution. Suppose the parameters are  = 5 and  = 0.1 (a) Compute E( ) and Var( ) (b) Compute P( > 120) (c) Compute P(110   130) (d) What is the value of median ductile strength? (e) If ten different samples of an alloy steel of this type were subjected to a strength test, how many would you expect to have strength at least 120? (f) If the smallest 5% of strength values were unacceptable, what would the minimum acceptable strength be? Lognormal Distribution - Example

Stracener_EMIS 7370/STAT 5340_Sum 08_ Lognormal Distribution –Example Solution a) b)

Stracener_EMIS 7370/STAT 5340_Sum 08_ Lognormal Distribution –Example Solution c) d)

Stracener_EMIS 7370/STAT 5340_Sum 08_ Lognormal Distribution –Example Solution e) Let Y=number of items tested that have strength of at least 120 Y=0,1,2,…,10

Stracener_EMIS 7370/STAT 5340_Sum 08_ Lognormal Distribution –Example Solution f) The value of x, say x ms, for which is determined as follows: