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MEGN 537 – Probabilistic Biomechanics
Ch.4 – Common Probability Distributions
Anthony J Petrella, PhD
Common Terms
• Random Variable: A numerical description of an experimental outcome. The domain (sometimes called the “range”) is the set of all possible values for the random variable
• Probability Distribution: A representation of all the possible values of a random variable and the corresponding probabilities.
Continuous and Discrete Probability Distributions
• Probability Distributions can be continuous or discrete based on the type of values contained within the domain of the random variable.
Normal or Gaussian Distribution
• Frequently, a stable, controlled process will produce a histogram that resembles the bell shaped curve also known as the Normal or Gaussian Distribution• The properties of the normal distribution make it a highly utilized
distribution in understanding, improving, and controlling processes
Common applications:Astronomical dataExam scoresHuman body temperatureHuman birth weightDimensional tolerancesFinancial portfolio managementEmployee performance
Normal Distribution
• Continuous Data• Typically 2 parameters
• Scale parameter = mean (mx)• Shape parameter = standard deviation (sx)
• CDF
x
21
exp2
1)x(f
2
x
x
xx
dx x
21
exp2
1)x(F
2
x
xx
xx
Distributions and Probability• Distributions can be linked to probability – making possible
predictions and evaluations of the likelihood of a particular occurrence
• In a normal distribution, the number of standard deviations from the mean tells us the percent distribution of the data and thus the probability of occurrence
Standard Normal Distribution
PDF CDF
0
0.1
0.2
0.3
0.4
0.5
-6.0 -4.0 -2.0 0.0 2.0 4.0 6.0x
f(x)
0
0.2
0.4
0.6
0.8
1
-6.0 -4.0 -2.0 0.0 2.0 4.0 6.0x
F(x
)
m = 0s = 1
Standard Normal Distribution
• Normal (m=0, s=1)• Standard normal variate
• (Note: Halder uses S)
• All normal distributions can be simply transformed to the standard normal distribution
• Probability )()( zFzyprobabilit
x
xxz
))a(z())b(z(dss21
exp)bxa(P 2)b(z
)a(z
Probability for other Sigma Values?
x
21
exp2
1)x(f
2
x
x
xx
• Suppose we want to calculate the amount of data included at X < 2.65s (Probability at 2.65s from the mean)• How will we figure out the area for such a particular standard
deviation measurement?
The probability density function is:
For given values of X, and s we could calculate the area under the curve, however, it would be unwise to go through this process every time we need to make a calculation
Solving for F(z)
• There is no closed form solution for the CDF of a normal distribution
• Common solution methods• Use a look-up table• Use a software package (Excel, SAS, etc.)• Perform numerical integration (e.g. apply
trapezoidal or Simpson’s 1/3 rule)
Experimental Data
• Fitting a distribution to the experimental data• Determine m and s • Use these as the distribution parameters
• Plot the raw data together with the normal curve representation and evaluate whether the distribution is normally distributed
Normal Distributions in Excel
General distributions• norm.dist(x,mean,stdev,cumulative) – returns
a probability at the specified value of the variable• cumulative = true (1) for CDF, cumulative = false (0) for PDF
• norm.inv(p,mean,stdev) – returns the value of the variable at the specified probability level
Standard normal distributions• norm.s.dist(z,cumulative) – returns probability• norm.s.inv(p) – returns the value of the std normal
variate, z
Means and Tails
• What aspects of data are most interesting from an engineering standpoint? Extreme conditions• Highest temperature or stress• Shortest life to failure
• Understanding the tails of a distribution can be critical to understanding performance• It is difficult to collect data in the tails distribution allows you to maximize dataRemember this is an assumption!
Lognormal Distribution
2xxx 2
1-ln))x(ln(E
• Natural log (ln) of the random variable has a normal distribution
• Determination of lognormal parameters from mean and standard deviation
Var(ln(x))
2
2 1lnx
xx
)xln(
21
expx2
1)x(f
2
x
x
xx
• Common applications:• Fatigue life to failure• Material Strength• Loading spectra
Lognormal Distribution
0
0.1
0.2
0.3
0.4
0.5
-2.0 0.0 2.0 4.0 6.0ln(x)
f(x
)
0
0.1
0.2
0.3
0.4
0.5
0 50 100 150 200 250x
f(x
)
m = 3s = 1
Lognormal Distribution
x
xxlnz
))a(z())b(z(
• Standard Normal Variate, z:
• Probability: dss21
exp)bxa(P 2)b(z
)a(z
Important Features
• From Haldar, p.71
• If X is a lognormal variable with parameters lx and zx, then ln(X) is normal with a mean of lx and a standard deviation of zx
• When COV, dx ≤ 0.3 zx ≈ dx,
Lognormal Distributions in Excel
General distributions• lognorm.dist(x,mean,stdev,cumulative) –
returns the probability • cumulative = true for CDF, cumulative = false for PDF
• lognorm.inv(p,mean,stdev) – returns the value of the variable
Transform with log and use same std. normal functions• norm.s.dist(z,cumulative) – returns probability• norm.s.inv(p) – returns the value of the std normal
variate, z