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

Investigation scour for underwater inspection of bridge, Define Investigati...

Define Investigation Scour for Underwater Inspection of Bridge? Investigation Scour is probably the most critical item to be aware of when performing an underwater inspection o

Surveying land, principle of working from whole to part

principle of working from whole to part

Strength of materials, An I-section beam 350*200mm2 has a web thickness of ...

An I-section beam 350*200mm2 has a web thickness of 12.5mm & a flange thickness of 25mm. It carries a shearing force of 200KN at a section. Sketch the shear stress distribution acr

Determine the allowable bearing pressure of the sand, Two load tests were c...

Two load tests were conducted at a site-one with a 0.5 m square test plate and the other with a 1.0 m square test plate. For a settlement of 25 mm, the loads were found to be 60 kN

Determine the grain exact gravity, (i) A dry soil has a void ratio of 0.65 ...

(i) A dry soil has a void ratio of 0.65 and its grain exact gravity is = 2.80. What is its unit weight? (ii) Water is added to the sample so that its degree of saturation is 60%

Explain about bentonite slurry and polymeric slurry, Q. Explain about Bento...

Q. Explain about Bentonite slurry and polymeric slurry? For the construction of diaphragm walls, bentonite slurry is generally used to form a filter cake on walls of trenches t

How to evaluate quality of bricks, Q. How to evaluate quality of Bricks? ...

Q. How to evaluate quality of Bricks? 1. Clour -Deep red, Carry red, Copper red, colour is indication of uniform burning and also suggests quality of clay used. Over brunt bric

Calculate annual interest rate which is compounded monthly, You wish to buy...

You wish to buy a house and you can afford to make a down payment of $50,000. Your monthly mortgage payment cannot exceed $2,000. If 30-year loans are available at 7.5% annual inte

Functions of different reinforcement in a typical pile cap, Functions of di...

Functions of different reinforcement in a typical pile cap Loads from columns transferring to pile cap persuade tensile forces at the bottom of cap. For example, by using trus

Show the application of concrete crash barriers, Q. Show the application of...

Q. Show the application of Concrete crash barriers? Concrete crash barriers aren't considered as the best barrier design due to the following reasons when compared with flexibl

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