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

Determine the coordinate for point r, For the project work points (x,y) sho...

For the project work points (x,y) shown in the figure the bearing of point R is due West. The coordinate for point R is most nearly:    Solution: Determine the length o

Open graded premix carpet, please tell me about gradation of open graded pr...

please tell me about gradation of open graded premix carpet

Statics - SFD & BMD Calculations, Could someone please help me with the cal...

Could someone please help me with the calculations of determining the sfd & bmd for cantilevers with udl''s, lvl''s and point loads? Also could I please have the same for ssb with

Continuous flow systems, Continuous Flow Systems: Transport by road, r...

Continuous Flow Systems: Transport by road, rail, air or ships takes place in vehicles which move along a path at intervals. As against this, continuous flow systems are those

Batching of concrete, Introdaction , volume batching , weighat batchin...

Introdaction , volume batching , weighat batching , mesaring of water

Mechanics of solids, Ask question #MinThe maximum instantaneous extension, ...

Ask question #MinThe maximum instantaneous extension, produced by an unknown falling weight through a height of 4 cm in a vertical bar of length 3 m and of cross sectional area 5 c

Heating season duration , Change the model to take into account the variati...

Change the model to take into account the variation in outdoor temperature for a "typical" day in the heating season based on the heat loss by the fabric and the heating of incomin

Calculate time taken for consolidation in actual layer, A clay layer 5 m th...

A clay layer 5 m thick has double drainage. It was consolidated under a load of 127.5 kN/m2. The load is enhanced to 197.5 kN/m 2 . The coefficient of volume compressibility is 5.7

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