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Probability Review

Probability Review. 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. Monte Carlo Method. Statistic Analysis: Have a set

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Probability Review

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  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 L3 / E W H3 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.

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