Compare the probabilities, Civil Engineering

Assignment Help:

Compare the probabilities:

Use the central limit theorem to compute P (¦ X - μ ¦ ≥ k), k = 1, 2,3,4,5 where X and μ are as defined in Example. Compare the probabilities with the probabilities computed in Example.

Solution:

We have np = 2, npq = 1, n = 4 and so

P(¦X - μ¦≥ k)=P (¦x-2¦√1 ≥ k/√1) = P(¦x-2¦≥ k)

+1-P(¦x-2¦< k) ∼ 1-[ Φ (k) - Φ (-k)]

=2(1- Φ(k))

Thus we tabulate the probabilities as below

 

496_Compare the probabilities.png

As expected, since the sample size 4 is not large sufficient, the difference between the exact and the approximate probabilities in Example are considerable. Thus the approximation will be better for larger values of n.

Approximations of equations are good for large n but better approximations are acquired by making some corrections, called the continuity corrections, for representing a discrete distribution through a continuous distribution. For the discrete binomial distribution P (x < X < x + 1) is equal to zero if x is an integer hut while this probability is approximated by the normal probability it is never equal to zero. This necessitates the continuity correction. A reasonable approach to get a better approximations to write P (X = x) as equal to

P (x-1/2

P ( xl ≤ X ≤ x2 ) as P( x1-1/2 < X ≤x2 + 1/2 ). This suggests that a better

approximation to binomial cdf is obtained by using

F(x) ∼ Φ  {(x+1/2 -np)/√(npq))} and

P(x1≤ X≤ x2) ∼ Φ  ((x2+1/2-np)/√(npq)) - Φ ((x1-1/2 -np)/√(npq))

alternatively of previous equations respectively. In fact, a theoretical justification of why this equation gives a better approximation than previous one can also be given. Thus, we do not go into the theoretical details.


Related Discussions:- Compare the probabilities

Explain the installation of shock transmission unit, Explain the Installati...

Explain the Installation of Shock Transmission Unit? It is not always necessary to install STUs while a structure is under construction but the holding down anchor bolts and th

Statics, Applications of statics

Applications of statics

Pavements with Bitumen Emulsion, What is the rate of application of Prime c...

What is the rate of application of Prime coat over WBM

Fillet welding, Strength of member when fillet welded

Strength of member when fillet welded

Distomat., write a note on distomat

write a note on distomat

Define the planning an underwater inspection, Define the Planning an Underw...

Define the Planning an Underwater Inspection? Planning for underwater inspection is important because of - Unknown factors which may be discovered during the diving - Th

Self compacting concrete, what will be the w/c ration and cement/fly ash ra...

what will be the w/c ration and cement/fly ash ration for the particular mix design

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