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

Functions and desirable characteristics of pavements, Functions and Desirab...

Functions and Desirable Characteristics of Pavements Function of Pavement A pavement is that part of a road on which vehicles travel. The pavement has the following f

RefConc Triangle Section''s bending stress, When I use allowable stress des...

When I use allowable stress design, how can I get depth of N.A and resultant force of Compression at triangle section? It''s Reinforced Concrete...

Determine the total lateral pressure on the wall, A masonry retaining wall ...

A masonry retaining wall is 1.5 m wide at the top, 3.5 m wide at the base and 6 m high. It is trapezoidal in section and has a vertical face on the earth side. The backfill is leve

Do you mean by accelerated aging, Q. Do you mean by Accelerated Aging? ...

Q. Do you mean by Accelerated Aging? Accelerated ageing tests are conducted at elevated temperature of 100 o C to know the variation in hardness, UTS and elongation at break.

Spatial structures loading, Spatial Structures Loading Dead load D...

Spatial Structures Loading Dead load Dead loads include the weight of permanent components such as elements and spheres weight, casing weight, false ceiling weight and uti

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

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

Determine the probable period of time, A clay stratum is 4.5 m thick and re...

A clay stratum is 4.5 m thick and rests on a rock surface. The coefficient of consolidation of a sample of this clay was found to be 4.5 × 10-8 m 2 /s in the laboratory. Verify pro

Ocf, location of hydrolic jump

location of hydrolic jump

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