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

Phase, Each physical homogeneous type (solid, liquid orvapour) is a phase. ...

Each physical homogeneous type (solid, liquid orvapour) is a phase. Two or three phases may coexist.  e.g. Ice and water at 0 o C.  Mixtures of gases (e.g. Air) can only be

Test leakage of constructed gravity pipelines, Question Have air test o...

Question Have air test or water tests be selected to test leakage of constructed gravity pipelines ? Answer For gravity pipes, air tests are carried out after completi

Ocf, location of hydrolic jump

location of hydrolic jump

Determine the types of untensioned reinforcement, Determine the types of un...

Determine the types of untensioned Reinforcement  Reinforcement used as untensioned steel should be one of the following types : (a)  Mild steel and medium tensile steel bar

Main principle of post-grouting for mini-piles, Question What is main p...

Question What is main principle of post-grouting for mini-piles? Answer During course of post-grouting , a tube with a hole at its base is lowered into pile and g

Fly levelling, we are given a project on field.i''d like to know about how...

we are given a project on field.i''d like to know about how to start n proceed vd fly levelling in the field

Large scale integration chips have between the components, Large scale int...

Large scale integration chips have between the  components? Answer : It is between 100 and 1000 components.

Coatings at the back faces of abutments, Q. Coatings at the back faces of a...

Q. Coatings at the back faces of abutments? There are many views on the necessity of application of protective coatings (can be in the form of two coats of paint) to the back f

Pavement design, differences between highway pavement and airfield pavement...

differences between highway pavement and airfield pavement

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