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

Types of cement, Types of Cement In addition to ordinary Portland ceme...

Types of Cement In addition to ordinary Portland cement there are various varieties of cement. Significant varieties are briefly explained below: (i) White Cement: The ce

Determine the ruthann graphical method, A retaining wall with a vertical ba...

A retaining wall with a vertical back 5 m high supports cohesion less backfill of unit weight of 19 kN/m 3 . The upper surface of the backfill rises at an angle of 10° with the hor

Design of the aeration basin based on solids retention time, Design of the ...

Design of the Aeration Basin Based on Solids Retention Time - sewer construction: You are provided the following information about a municipal wastewater treatment plant. That

Bricks, charactor of fourth class brick

charactor of fourth class brick

Bolder, what is a bolder?

what is a bolder?

Screw pumps and pump trough, Question In structure of pump troughs for ...

Question In structure of pump troughs for accommodation of screw pumps, what is major construction process to ensure close contact between screw pumps and pump trough ? An

Torpedo splice triple wedges, Torpedo splice triple wedges In a Torpedo...

Torpedo splice triple wedges In a Torpedo splice triple wedges, shown in Figure, are provided to firmly hold tendon wires. Whole assembly of the splice is housed in a sleeve. L

Calculate the tension force in sling - lifting load, Calculate the Tension ...

Calculate the Tension Force in Sling The rigging configuration shown below will be used to lift a bridge girder using a 200-ton capacity luffing jib crane onto the foundation a

Calculate uniform annual cost for maintenance and repair, A contractor purc...

A contractor purchased a Caterpillar D-13 dozer for $100,000 and owned it for five years. The annual cost for maintenance and repair is provided in the Table. With annual interest

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