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MEGN 537 – Probabilistic Biomechanics Ch.4 – Common Probability Distributions

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

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MEGN 537 – Probabilistic Biomechanics Ch.4 – Common Probability Distributions

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  1. MEGN 537 – Probabilistic BiomechanicsCh.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 • 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. 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? Probability for other Sigma Values? 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 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

  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 dataRemember this is an assumption!

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

  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 lxand zx, then ln(X) is normal with a mean of lxand a standard deviation of zx • When COV, dx ≤ 0.3zx≈ dx,

  22. 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

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