Compare the probabilities, Civil Engineering

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.

Posted Date: 1/31/2013 5:37:31 AM | Location : United States







Related Discussions:- Compare the probabilities, Assignment Help, Ask Question on Compare the probabilities, Get Answer, Expert's Help, Compare the probabilities Discussions

Write discussion on Compare the probabilities
Your posts are moderated
Related Questions
Q. Explain Remote sensing technique? Remote sensing is a science or an art of obtaining the information about an object, area or a phenomenon through analysis of data acquired

Q. Core-barrel samplers: single tube sampler vs. double tube sampler vs. triple tube sampler Core barrel samplers are initially designed to sample rock. In single tube sampler

Explain the following: (a) Sea breeze and Land breeze (b) What is Thermal comfort (c) What is Global wind pattern and Coriolis effect (d) How wind affects the tempe

Soil Compaction: Soils used in road construction need to be thoroughly rolled and compacted. Compaction of soil is needed for the following reasons. (a) A well compacted s

Bricks may be broadly classified as following: Paving bricks Building bricks Special bricks A.   Fire bricks B.   Building Bricks: These bricks are utilized fo

A storm occurs with rain of varying intensity falling over a commercial area. The area has a combined sewer system. Two interceptor sewers direct flow to the wastewater treatment p

Question What is function of shear keys in designing of retaining walls ? Answer In formative external constancy of retaining walls, failure modes similar to bearing failure, s

defference between direct and indirect ranging..

Calculate the Benefit-Cost Ratio for Municipal Government Project With interest at 10%, the benefit-cost ratio for this municipal government project is most nearly:   Initia

Q. Show the Overlays on concrete bridge deck? After years of servicing, some overlays can be applied on the top surface of bridges. Overlays on concrete bridge decks achieve th