Univariate normal distribution, Civil Engineering

Assignment Help:

Univariate normal distribution:

Let X = (XI, X2, ..., Xn,) has the multivariate normal distribution (5.26) of Section 5.4. Show that Y = a X ' follows a univariate normal distribution where a = ( a1 a2 ..., an, ). Also find the mean and the variance of Y.

Solution:

We have

My(t)= E (etY)= E(etaX') = E(eτX')

where τ = ta = (ta1, ta2,..., tan ).

From Example 3 of Section 5.4, we have

MX (t) = E(e tX') = e τμ' +1/2(t∑ t')

Hence

My(t) = MX(τ) = eτμ' - ½ (τ∑τ') =e t(a μ')-1/2 t2(a∑a')

=e -1/2 t2 σ2

Where

1083_Univariate normal distribution.png

842_Univariate normal distribution3.png

Hence

MX(t) = e tμ+1/2 (t2σ2)

in other words Y - N ( μ, σ2 ) where

682_Univariate normal distribution2.png

842_Univariate normal distribution3.png


Related Discussions:- Univariate normal distribution

Failure analysis in designing of a structure, Modes of failure • For sa...

Modes of failure • For safe design, a knowledge of the possible modes of failure is also necessary, e.g. failure of the structure could occur due to - Tensile fracture due t

Sub-base layer be served as a drainage layer , Question Would a sub-bas...

Question Would a sub-base layer be served as a drainage layer to take away water from pavement? Answer As well providing load distribution in post-construction stage a

Calculate the surface dimensions of sedimentation tank, Calculate the surfa...

Calculate the surface dimensions of Sedimentation Tank A treatment plant upgrade calls for design of sedimentation basins to handle a flow rate of 8 MGD. The basins are for a T

Track design for line speed improvements, The objective of this module is:-...

The objective of this module is:- For the student to identify the track design issues and there relationships that will need to be considered in the development of a feasibil

Premix road, plz tell me the ratio of sand,aggregate,and asphalt.pl tell me...

plz tell me the ratio of sand,aggregate,and asphalt.pl tell me the mm of aggregate

Building technology 1, what are the defects in timber due to natural forces...

what are the defects in timber due to natural forces?

Aslar masonry, different types of ashlar masonry

different types of ashlar masonry

Write Your Message!

Captcha
Free Assignment Quote

Assured A++ Grade

Get guaranteed satisfaction & time on delivery in every assignment order you paid with us! We ensure premium quality solution document along with free turntin report!

All rights reserved! Copyrights ©2019-2020 ExpertsMind IT Educational Pvt Ltd