(a) Determine the Nataf model for the joint PDF fxx, (xx) of the basic (physical) random variables X₁ and X, with marginal PDF's fx(x)=e, 0≤x (Exponential distribution) fx₁ (x2)=x2e-0.5x, 0≤x (Rayleigh distribution) and correlation coefficient Pxx=0.50 Note: Use Table 6 of paper by Liu and Der Kiureghian, 1986. (b) Generate a 3D surface plot and contour plot of the joint PDF fxx, (x,x) using Matlab or any other software of your choice. (c) What is the standard deviation of X2? (d) Construct a transformation from the physical X space (defined by random variables X, and X,) to the standard normal U space (defined by the statistically independent standard normal random variables (U, and U₂), i.e., U=T(X). Also describe the inverse transform X=T(U) and the Jacobian matrices J = ди θα and Ju Ox ди (e) According to the inverse transformation X = T¹ (U) and using Matlab, generate 1,000 samples from the Nataf joint PDF fxx, (x1,x2) derived in part (a). Start by generating samples of U using a specified seed number and then follow the transformations from U to Z and from Z to X (see lecture notes on Nataf Model). Generate a scatter diagram of the 1,000 samples superimposed on a contour plot of the Nataf joint PDF fxx, (xx). Repeat the above for a different seed number.
(a) Determine the Nataf model for the joint PDF fxx, (xx) of the basic (physical) random variables X₁ and X, with marginal PDF's fx(x)=e, 0≤x (Exponential distribution) fx₁ (x2)=x2e-0.5x, 0≤x (Rayleigh distribution) and correlation coefficient Pxx=0.50 Note: Use Table 6 of paper by Liu and Der Kiureghian, 1986. (b) Generate a 3D surface plot and contour plot of the joint PDF fxx, (x,x) using Matlab or any other software of your choice. (c) What is the standard deviation of X2? (d) Construct a transformation from the physical X space (defined by random variables X, and X,) to the standard normal U space (defined by the statistically independent standard normal random variables (U, and U₂), i.e., U=T(X). Also describe the inverse transform X=T(U) and the Jacobian matrices J = ди θα and Ju Ox ди (e) According to the inverse transformation X = T¹ (U) and using Matlab, generate 1,000 samples from the Nataf joint PDF fxx, (x1,x2) derived in part (a). Start by generating samples of U using a specified seed number and then follow the transformations from U to Z and from Z to X (see lecture notes on Nataf Model). Generate a scatter diagram of the 1,000 samples superimposed on a contour plot of the Nataf joint PDF fxx, (xx). Repeat the above for a different seed number.
Engineering Fundamentals: An Introduction to Engineering (MindTap Course List)
5th Edition
ISBN:9781305084766
Author:Saeed Moaveni
Publisher:Saeed Moaveni
Chapter15: Computational Engineering Tools Matlab
Section15.5: Symbolic Mathematics With Matlab
Problem 1BYG
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Transcribed Image Text:(a) Determine the Nataf model for the joint PDF fxx, (xx) of the basic (physical) random variables X₁
and X, with marginal PDF's
fx(x)=e, 0≤x
(Exponential distribution)
fx₁ (x2)=x2e-0.5x, 0≤x (Rayleigh distribution)
and correlation coefficient
Pxx=0.50
Note: Use Table 6 of paper by Liu and Der Kiureghian, 1986.
(b) Generate a 3D surface plot and contour plot of the joint PDF fxx, (x,x) using Matlab or any other
software of your choice.
(c) What is the standard deviation of X2?
(d) Construct a transformation from the physical X space (defined by random variables X, and X,) to the
standard normal U space (defined by the statistically independent standard normal random variables
(U, and U₂), i.e., U=T(X). Also describe the inverse transform X=T(U) and the Jacobian
matrices J
=
ди
θα
and Ju
Ox
ди
(e) According to the inverse transformation X = T¹ (U) and using Matlab, generate 1,000 samples from
the Nataf joint PDF fxx, (x1,x2) derived in part (a). Start by generating samples of U using a specified
seed number and then follow the transformations from U to Z and from Z to X (see lecture notes
on Nataf Model). Generate a scatter diagram of the 1,000 samples superimposed on a contour plot of
the Nataf joint PDF fxx, (xx).
Repeat the above for a different seed number.
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