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

Sand cone test – suitable for all soils?, Q. Sand cone test – suitable for ...

Q. Sand cone test – suitable for all soils? Sand cone (replacement) test is generally carried out to determine the in-situ and compacted density of soils. This testing method i

Boundary ratio of weight of hammer to weight of pile, Question If n a n...

Question If n a number of codes, they boundary ratio of weight of hammer to weight of pile for pile driving. What will be the key reason for this? Answer As a hamm

Deformation equation for settlement, Q. Deformation Equation for settlement...

Q. Deformation Equation for settlement? When a load is applied on an elastic body, it results in strain. The summation of strain over a length is deformation or settlement. In

Void ratio, a soil sample is 38mm in diameter and 76mm and in its natural c...

a soil sample is 38mm in diameter and 76mm and in its natural condition weighs 168g.when dried completely in an oven weighs 13o.5g.the value of Gs is 2.73.what is the degree of sat

Test on specially moulded test piece, Test on specially moulded test piece ...

Test on specially moulded test piece The test piece shall be moulded by the manufacturer with identical compound and under identical vulcanizing conditions as used in manufact

Backward and forward curved vanes in pumps, Backward curved vanes vs. forwa...

Backward curved vanes vs. forward curved vanes in pumps  Power of a pump is related to discharge as follows: Power = k 1 Q + (k 2 Q 2 / tan A) Where K 1 and K 2 are

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