MEGN 537 – Probabilistic Biomechanics Ch 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 data Exam scores Human body temperature Human birth weight Dimensional tolerances Financial portfolio management Employee performance

Normal Distribution Continuous Data Typically 2 parameters PDF CDF Scale parameter = mean (mx) Shape parameter = standard deviation (sx) PDF CDF

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 CDF PDF m = 0 s = 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

Probability for other Sigma Values? 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

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 PDF value or CDF value at x 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 PDF or CDF 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 data Remember this is an assumption!

Lognormal Distribution Natural log (ln) of the random variable has a normal distribution Determination of lognormal parameters from mean and standard deviation

Lognormal Distribution Common applications: Fatigue life to failure Material Strength Loading spectra m = 3 s = 1

Lognormal Distribution where l=scale and z = shape

Lognormal Distribution Standard Normal Variate, z: Probability:

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 PDF or CDF value 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 PDF or CDF norm.s.inv(p) – returns the value of the std normal variate, z

Continuous Data Normal Distribution - infinity to + infinity Lognormal Distribution 0 to + infinity

Beta Distribution Bounded: where B(a,b) is the “beta function” X PDF CDF X Images: http://en.wikipedia.org/wiki/Beta_distribution

Truncated Normal PDF: where is the PDF of the normal curve, and is the CDF of the normal curve Applicable for tolerance ranges on dimensions

Uniform Distribution PDF: Applicable for tolerance ranges on dimensions, temporal variations PDF L U CDF L U 1.0

Mean and COV Expressions for E(X) and COV(X) for different distributions