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)

• PDF

• CDF

x

21

exp2

1)x(f

2

x

x

xx

dx x

21

exp2

1)x(F

2

x

xx

xx

Normal Distribution

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

The Standard Normal Distribution

Negative z Values

)(1)( zz

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

where l=scale and z = shape

)xln(

21

expx2

1)x(f

2

x

x

xx

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