OCE301 Part VI: Probability Theory lecture 3

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OCE301 Part VI Probability Theorylecture 3
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Rayleigh Wave Height Model

• Longuet-Higgins (1952) demonstrated that the
  Rayleigh distribution is applicable to the
  heights of waves in the sea.

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Rayleigh Distribution
cdf
a monotonically increasing function
pdf
poe
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Root-Mean-Square Value
pdf
(mean-square value)
second moment
notation for the rms of H
root-mean-square (rms) value
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Rayleigh Distribution in RMS Value
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Matlab Display Rayleigh
hrms1 h00.014 psd 2h/hrms2 .
exp(-(h/hrms).2) cdf 1 - exp(-(h/hrms).2) su
bplot(211) plot(h,psd) grid subplot(212)
plot(h,cdf,'r-') grid
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Rayleigh Distribution in Mean Value
(drop subscript H from mH for simplicity)
pdf in terms of mean value
(change of parameter)
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Change of Variable
change of variable
normalized wave height
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Distributions of Two Random Variables
2-D (cumulative) distribution function
Discrete random variables
Total probability
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Continuous 2-D Distributions
Continuous random variables
Probability in a rectangular
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Marginal Distribution
Marginal distribution function of X in (X,Y)
Marginal pdf of X in (X,Y)
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Set 22.9 (prob. 15)
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Expectation of a Function of Two Random Variables
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Independence of Random Variables
Two random variables, X and Y, are independent if
This can be extended to n-dimensional random
variables
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Multiplication of Means
If X and Y are independent
can be extended to n random variables
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Variance of ZXY
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Correlation Coefficient
correlation coefficient of X and Y
dimensionless
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Uncorrelated vs. Independent
independent
uncorrelated
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Problem set 22.9 (10)
If the weight of certain (empty) containers has
5 lb and standard deviation 0.2 lb, and if the
filling of the containers has mean weight 100
lb and standard deviation 0.5 lb, what are the
mean weight and the standard deviation of filled
containers?
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Probability of Exceedance
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Wave Height of Specific POE
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Graphic Demonstration on H1/n
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Compute H1/n from Rayleigh Model
lower bound from Rayleigh model
Rayleigh pdf
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Mathematical Relationship between H1/n and Hrms
Following Rayleigh model for wave height
erfc(x) is the complementary error function
defined as
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Numerical Relationship among Representative Wave
Heights