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Class 03 Random variables

# Class 03 Random variables - maximum size of inclusions in...

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T. R. Willemain MAU Spring 06 1 Agenda Definitions: Experiments and random variables Types of RVs: Continuous vs discrete Events and their probabilities Probability distributions: Pdf’s and Cdf’s Expectations: Mean and variance Using Minitab and Maple

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T. R. Willemain MAU Spring 06 2 Events and their probabilities “Event” = A set of possible outcomes 0 ≤ P(event) ≤ 1 Ex: P(snowfall between 1 and 2 inches) Mutually exclusive events E 1 , E 2 – P(E1 or E2) = P(E 1 )+P( E 2 ) Ex: P(snowfall is 0-1” or >9”) =P(0-1”)+P(>9”) Complementary events E, E’ P(E or E’) = 1 Ex: P(Live) = 1-P(Die)
T. R. Willemain MAU Spring 06 3 Probability distributions Let f(X) = pdf of X Let F(X) = cdf of X = ∫f(U)dU, from -∞ to X Let event of interest be: a < X < b P(a < X < b) = ∫f(X)dX, from a to b P(a < X < b) = F(b)-F(a)

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T. R. Willemain MAU Spring 06 4 CDF PDF
T. R. Willemain MAU Spring 06 5 The precision of methods using the statistics of extremes for the estimation of the

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Unformatted text preview: maximum size of inclusions in clean steels. C. W. Anderson, G. Shi, H. V. Atkinson, and C. M. Sellars Acta Materialia V olume 48, Issue 17 , November 2000, Pages 4235-4246 T. R. Willemain MAU Spring 06 6 Histogram is empirical estimate of PDF T. R. Willemain MAU Spring 06 7 T. R. Willemain MAU Spring 06 8 T. R. Willemain MAU Spring 06 9 T. R. Willemain MAU Spring 06 10 Expectation • f(X) = pdf • F(X) = cdf = integral of f() from -∞ to X • Expectations are shorthand summaries of distributions • E[X] = ∫Xf(X)dX= μ • Var[X] = ∫(X-μ) 2 f(X)dX = σ 2 = E[(X- μ) 2 ] • Stdev[X] = √σ 2 = σ T. R. Willemain MAU Spring 06 11 using Minitab and Maple T. R. Willemain MAU Spring 06 12 T. R. Willemain MAU Spring 06 13 T. R. Willemain MAU Spring 06 14 T. R. Willemain MAU Spring 06 15 T. R. Willemain MAU Spring 06 16 T. R. Willemain MAU Spring 06 17 Check that have legitimate pdf that integrates to 1....
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