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Use of Normal Distribution
ashokkr(13160)
27 September 2015
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Introduction
NORMAL DISTRIBUTION:-Normal distributions are a family of distributions that have the same general
shape. They are symmetric with scores more concentrated in the middle than in the tails. Normal distributions are sometimes described as bell shaped. The simplest case of normal distribution, known as the
Standard Normal Distribution, has expected value zero and variance one. This is written as N (0, 1). If we
know the value of mean and varience of any distribution then we find prob of this distribution easily using
N(0,1) using table of this distribution.
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Characteristics of N(0,1)
• The area under the curve is 1
• It is a continuous distribution
• It is symmetrical about the mean. Each half of the distribution is a mirror image of the other half.
• It is asymptotic to the horizontal axis.
Examples:• Many things are normally distributed, or very close to it. For example, height and intelligence are approximately normally distributed; measurement errors also often have a normal distribution.
• The normal distribution is easy to work with mathematically. In many practical cases, the methods developed using normal theory work quite well even when the distribution is not normal.
• There is a very strong connection between the size of a sample N and the extent to which a sampling distribution approaches the normal form. Many sampling distributions based on large N can be approximated
by the normal distribution even though the population distribution itself is not normal.
• The normal distribution is the most used statistical distribution. The principal reasons are: normality
arises naturally in many physical, biological, and social measurement situations.
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Test strengths and weaknesses
The strengths of normal distribution are:
• probably the most widely known and used of all distributions.
• infinitely divisible probability distributions.
• strictly stable probability distributions.
The weakness of normal distributions is for reliability calculations. In this case, using the normal distribution
starts at negative infinity. This case is able to result in negative values for some of the results
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Statistical formula
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P(x) =e−(x−µ) /2σ σ√12π
W here : −
µ = mean of x.
σ=varience of x.
π =3.14159.....
e=2.7182.....
• We first convert the problem into an equivalent one dealing with a normal variable measured in standardized deviation units, called a standardized normal variable. To do this,
if X ∼ N (µ, σ), then Z = X- µ/ σ ∼ N (0, 1)
• A table of standardized normal values (from the table in the first appendix of this paper)
can then be used to obtain an answer in terms of the converted problem.
• If necessary, we can then convert back to the original units of measurement. To do this,
simply note that, if we take the formula for Z, multiply both sides by , and then add to
both sides, we get X= Z σ + µ
• The interpretation of Z values is straightforward. Since σ= 1, if Z = 2, the corresponding
X value is exactly 2 standard deviations above the mean. If Z = -1, the corresponding X
value is one standard deviation below the mean. If Z = 0, X = the mean, i.e. µ.
References
[1] Cristiano Maio1 and Cliff Schexnayder ’Fulbright Graduate Student and 2Eminent Scholar,
Del E. Webb School of Construction, Box 870204, Ariz. State Univ., Tempe, AZ 852870204, USA. [email protected]
[2] AbouRizk, S.M., and Halpin, D.W. (1990). ”Probabilistic simulation studies for repetitive
construction processes.” J. of Constr. Engrg. and Mgrnt., ASCE, 116(4), 575-594
[3] aw, A.M. and Kelton, W.D. (1991). Simulation modeling and analysis. McGraw-Hill, NY
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