Special Continuous Probability Distributions Normal Distributions Lognormal Distributions

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*Special Continuous Probability DistributionsNormal DistributionsLognormal Distributions
Dr. Jerrell T. Stracener, SAE FellowLeadership in EngineeringEMIS 7370/5370 STAT 5340 : PROBABILITY AND STATISTICS FOR SCIENTISTS AND ENGINEERS Systems Engineering ProgramDepartment of Engineering Management, Information and Systems

*A random variable X is said to have a normal (orGaussian) distribution with parameters and ,where  < < and > 0, with probability density function
for  < x < Normal Distribution

*
the effects of and
Properties of the Normal Model

*
Mean or expected value ofMean = E(X) =
Median value of
X0.5 =
Standard deviationNormal Distribution

*
Standard Normal Distribution
If ~ N(, )
and if
then Z ~ N(0, 1).
A normal distribution with = 0 and = 1, is calledthe standard normal distribution.Normal Distribution
 *x0zf(x)f(z)P (X

*Normal Distribution

*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) = 0.05Normal Distribution  Example

*
a. P( < 105.3) =
= P( < 0.53)= F(0.53)= 0.7019100x0zf(x)f(z)105.30.53Normal Distribution Example Solution

*
b. P( 91.7) =
= P( 0.83) = 1  P( < 0.83) = 1 F(0.83) = 1  0.2033 = 0.7967100x0zf(x)f(z)91.70.83Normal Distribution Example Solution

*
c. P(87.1 < 115.7)= F(115.7)  F(87.1)
= P(1.29 < Z < 1.57)= F(1.57)  F(1.29)= 0.9418  0.0985 = 0.8433100xf(x)87.1115.70xf(x)1.291.57Normal Distribution Example Solution

*100x0zf(x)f(z)0.050.051.64116.4Normal Distribution Example Solution

*
(d)P( x) = 0.05P( z) = 0.05implies that z = 1.64 P( x) =
therefore
x  100 = 16.4x = 116.4Normal Distribution Example Solution

*The time it takes a driver to react to the brake lightson a decelerating vehicle is critical in helping toavoid rearend collisions. The article FastRise BrakeLamp as a CollisionPrevention Device suggests that reaction time for an intraffic response to abrake signal from standard brake lights can be modeled with a normal distribution having meanvalue 1.25 sec and standard deviation 0.46 sec.What is the probability that reaction time is between1.00 and 1.75 seconds? If we view 2 seconds as acritically long reaction time, what is the probabilitythat actual reaction time will exceed this value?Normal Distribution Example Solution

*Normal Distribution Example Solution

*Normal Distribution Example Solution

*Lognormal Distribution

*
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
Lognormal Distribution

*
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

*Mean or Expected Value Standard Deviation MedianLognormal Distribution  Properties

*A theoretical justification based on a certain materialfailure 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

*Lognormal Distribution Example Solutiona) b)

*Lognormal Distribution Example Solutionc)d)

*Lognormal Distribution Example Solutione)Let Y=number of items tested that have strength of at least 120y=0,1,2,,10

*Lognormal Distribution Example Solutionf) The value of x, say xms, for which is determined as follows: