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

Exponential distribution - poisson distribution function, Exponential distr...

Exponential distribution - Poisson distribution function: Let X follow exponential distribution with parameter θ. Find the pdf of Y = X 1/β where  β( ≥ 1 ) is a constant.

Consultant , how to convert a varying distributed load to a uniformly distr...

how to convert a varying distributed load to a uniformly distributed load

Explain the hydraulic gradient, Explain the Hydraulic Gradient A measur...

Explain the Hydraulic Gradient A measurement of the movement of water over distances. Change in hydraulic head from one monitoring point to another.  i = ΔH/L  units are typ

Retaining walls, Retaining Walls: Needed to retain the fill portion, a...

Retaining Walls: Needed to retain the fill portion, and generally constructed in dry stone masonry with pucca bunds (Figure). Figure: Typical Design for a Retaining

Concrete structure be entirely free of expansion joints, Question Would a c...

Question Would a concrete structure be entirely free of expansion joints and contraction ? Answer Consider that concrete structure is not subject to problem of disparity settlem

Application of levelling, how do I draw sections(longitudinal and cross sec...

how do I draw sections(longitudinal and cross sections)?

Linear algebra concept, Scenario: You are working for a structural engineer...

Scenario: You are working for a structural engineering firm and you have been assigned the task of computing the code checks on a new structural design. Though another commercial p

BUILDING CONSTRUCTION, GIVE IN DETAIL THE TREATMENT OF TOP SURFACE OF EARTH...

GIVE IN DETAIL THE TREATMENT OF TOP SURFACE OF EARTH FILLING..

Compute the porosity of the initial soil sample, A completely saturated sam...

A completely saturated sample of clay has a volume of 31.25 cm 3 and a weight of 0.5866 N. The same sample after drying has a volume of 23.92 cm 3 and a weight of 0.4281 N. Compu

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