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

Coating on concrete – complete im-permeability to moisture, Q. Coating on c...

Q. Coating on concrete – complete im-permeability to moisture? In designing protective coating on concrete structures, stoppage of water ingress by the coating is generally req

Determine the statements about unit area or borrow pit, Determine the State...

Determine the Statements about Unit Area or Borrow Pit? Ans: I.  Sometimes called Average Depth II. Works well for volumes for building sites and surface mines III. Needs

Motion in one dimension, An F-16 lands on aircraft carrier and its velocity...

An F-16 lands on aircraft carrier and its velocity is 63 m/sn. If its decceleration is assumed to be constant and it must stop in 2.0 sec. a) What is final position of plane if i

Surveying, what are tie lines? explain in detail

what are tie lines? explain in detail

Designing of access ramps for drainage channels, Question Designing of ...

Question Designing of access ramps for drainage channels, why must direction of access ramps is sloping down on the way to downstream ? Answer In designing of access

Define facilities that discharge wastewater to a potw, Facilities that disc...

Facilities that discharge wastewater to a publicly owned treatment works? Facilities that discharge wastewater to a POTW (publicly owned treatment works) are known as subject t

Base isolation, why base isolation is not good for tall building

why base isolation is not good for tall building

Strength of Materials, Describe the process of yielding and state the effec...

Describe the process of yielding and state the effect work hardening has on the yield point of mild steel using suitable sketches where appropriate.

Laplace transform, Using Laplace transfom, the following differential equat...

Using Laplace transfom, the following differential equation d y dy dt reduces to the algebraic equation of the form Resolve y(s) into partial fractions

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