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Unformatted text preview: nt and has a probability of occurrence of . The probabilities of the two events are given by the Bernoulli distribution.
for . Find the variance.
(Kottegoda and Rosso, 2008) Solution. If we take the second moment about the origin, then using the expression for
variance,
, we get: Let , now as , the variance is given by, 4. Expansion of Functions of Random Variable
The function of random variable,
value, can be expanded in a Taylor series about the mean . 1
dg 1
d 2g ( X X )2 Y g ( X ) ( X X )
1!
dX 2!
dX 2
where derivatives are evaluated at . If the series is truncated at linear terms, then the firstorder approximate mean and variance of
are obtained. The variance of function of random variable,
It should be noted that, if the function, is approximately linear for the entire range of value , then above two equations will yield good approximation of exact moments.
Problem 3. The maximum impact pressure, of ocean waves on coastal structures is determined by:
where, =density of water; =length of hypothetical piston; =thickness of air cushion and =horizontal velocity of the advancing wave.
Suppose that the mean crest velocity is
The density of sea water is about ft/s with a coefficient of variation, slugs/cu.ft and the ratio of . . Determine the mean and standard deviation of the peak impact pressure.
Solution. We have
So, Similarly, So, 5. Moment Generating Functions for Derived Random Variable
A random variable that is a function of other random variables and its probability distribution
are also defined as a derived variable and a derived variable, respectively. The determination
of the probability distribution of a derived variable from those of t...
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This note was uploaded on 03/18/2014 for the course CE 5730 taught by Professor Dr.rajibmaity during the Spring '13 term at Indian Institute of Technology, Kharagpur.
 Spring '13
 Dr.RajibMaity
 Civil Engineering

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