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

Truss, modified pratt truss with detailing in 3D

modified pratt truss with detailing in 3D

Fluid Mechanics, A 4-m-long, 6-cm-diameter cylinder placed normal to the fl...

A 4-m-long, 6-cm-diameter cylinder placed normal to the flow is used to measure the velocity of an air stream. If a force of 50 N is measured, the air velocity is nearest (for air

Accuracy and range - angle measurement, Accuracy and Range - Angle measurem...

Accuracy and Range - Angle measurement: Accuracy and Range:-Angle measurement:-Most modern total station instruments measure angles by means of electro-optical scanning of ext

Measure the cohesion of internal friction of the soil, A cylindrical specim...

A cylindrical specimen of a saturated soil fails under an axial stress 150 kN/m 2 in an unconfined compression test. The failure plane makes an angle of 52° with the horizontal.

Majority carriers are hols in, Majority carriers are hols in which type? ...

Majority carriers are hols in which type? Answer: In P-type semiconductors hols are majority carriers .

Concrete structural members tensile stresses, Concrete structural members t...

Concrete structural members tensile stresses In concrete structural members tensile stresses act in a longitudinal direction. This is why, stretched tendons (hereafter, called

State the laws governing compaction of cohesion less soils, a) What are th...

a) What are the laws governing compaction of (i) cohesion less soils like sand and (ii) moderately cohesive soils like sandy clay ? b) To what compaction pressure does the

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