1. Probability Review

2. Probability Probability = mathematic interpretation of uncertainty –Uncertainty plays a major role in engineering decision making. Set = collection of: –Items –Events –Occurrences Distribution = behavior of a set

3. Monte Carlo Method Statistic Analysis: –Have a set –Derive a distribution Monte Carlo Method: –Have a distribution –Construct a model set

4. Example 1 Deterministic calculation of deflection for a cantilever beam with quadratic cross section: Deflection = 4 F L 3 / E W H 3 L = length of beam W = width of beam H = height of beam I = area moment of inertia E = Young’s modulus F = applied downward force

5. Example 1 Stochastic Calculation: Deflection = 4FL^3/EWH^3 –Symbols (physical parameters) represent distributions (expressed in MATLAB as vectors). –Vectors (distributions) should: have the same number of elements be randomly constructed according to preset rules regarding each quantity.

6. Common Distributions Uniform: Constant probability over a range of values. Useful for round-off errors Normal/Gaussian: Bell curve. Useful for large samples of random occurrences such as height.

7. Common Distributions Gamma: Only defined for positive x Useful for time dependant events, arrivals, etc. Exponential: A form of the Gamma, memory- less (events do not affect following occurrences) Weibull: A good representation of the frequency of failure for many types of equipment

8. Deterministic v. Stochastic Results

9. Programs Matlab –More than Matrices –Useful tool for Monte Carlo Modeling Excel –Used to process results of Matlab models

10. Useful Commands in Matlab R = unifrnd(A,B,m,n) generates uniform random numbers with parameters A and B, where scalars m and n are the row and column dimensions of R. R = normrnd(MU,SIGMA,m,n) generates normal random numbers with parameters MU and SIGMA, where scalars m and n are the row and column dimensions of R. R = gamrnd(A,B,m,n) generates gamma random numbers with parameters A and B, where scalars m and n are the row and column dimensions of R. R = exprnd(MU,m,n) generates exponential random numbers with mean MU, where scalars m and n are the row and column dimensions of R. R = wblrnd(A,B,m,n) generates Weibull random numbers with parameters A and B, where scalars m and n are the row and column dimensions of R.