1. MEGN 537 – Probabilistic Biomechanics Ch
MEGN 537 – Probabilistic Biomechanics Ch.4 – Common Probability Distributions
Anthony J Petrella, PhD

2. 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.

3. 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.

4. 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

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

6. Normal Distribution

7. 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

8. Standard Normal Distribution
CDF
PDF
m = 0
s = 1

9. 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

10. 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

11. The Standard Normal Distribution

12. Negative z Values

13. 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)

14. 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

15. 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

16. 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!

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

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

19. Lognormal Distribution
where l=scale and z = shape

20. Lognormal Distribution
Standard Normal Variate, z:
Probability:

21. 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,

22. 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

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

24. Beta Distribution Bounded: where B(a,b) is the “beta function” X PDF
CDF X
Images:

25. 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

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

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