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NORMAL OR GAUSSIAN DISTRIBUTION Chapter 5
General Normal Distribution Two parameter distribution with a pdf given by:
Estimators for 1 and 2 Using MOM or Maximum Likelihood to estimate 1 and 2 :
Properties of Normal Distribution Bell-shaped Continuous Symmetrical about the mean
Properties of the Normal Distribution Mean and variance are sometimes referred to as location and scale parameters. Varying 2 while holding constant changes the scale but not the location of the distribution. Varying while holding 2 constant changes the location of the distribution but not the scale.
Notation Common to denote a random variable that is normally distributed with a mean, and variance 2 as N(, 2 ).
Reproductive Properties of the Normal Distribution If X is a r.v., and, then the distribution of Y can be shown to be If X i for i = 1,2,…n are independently and normally distributed with mean i and variance i 2, then is normally distributed with:
Example If x i is a random observation N(, 2 ) what is the distribution of ?
Standard Normal Distribution The cdf for the normal distribution is: To evaluate we must use a linear transformation so that the random variable will be N(0,1).
Standard Normal Distribution When we make this transformation, Z is said to be standardized and N(0,1) is the standard normal distribution. The standard normal pdf and cdf are:
Using the Standard Normal Distribution Probabilities for the standard normal distribution are widely tabulated. Most tables make use of symmetry and show only positive values of Z. Tables may show: Your book has standard normal distribution in Appendix A.12. Gives prob(Z
Standard Normal Distribution The table shows that 68.26% of normal distribution is within 1 of the mean % of normal distribution is within 2 of the mean % of normal distribution is within 3 of the mean. These are the 1, 2, and 3 sigma bounds of the normal distribution.
Standard Normal Distribution Only 0.26% of the area of the normal distribution lies outside the 3 rd sigma bound. The probability of a value less than is only Since many of our r.v. are bounded at the left by x=0, this gives us justification for using the normal distribution in instances where is greater than because the chance that X < 0 is many times (but not always) negligible.
Example NO3-N concentrations (ppm) in a local well are N(3,0.2). a) What is the probability that a sample drawn at random will be between 2 and 7 ppm. b) What is the probability that a sample will have a concentration exceeding 10 ppm?
Central Limit Theorem If S n is the sum of n independently and identically distributed random variables X i each having a mean,, and variance, 2, then in the limit as n approaches infinity, the distribution of S n approaches a normal distribution with mean n and variance n 2.
Normal Approximations for Other Distributions Under certain conditions the normal distribution can be a good approximation to several other distributions, both discrete and continuous. Generally approximations are good in the center of the distribution with accuracy dropping off in the tails. C.L.T. provides the mechanism by which the normal distribution becomes an approximation for several other distributions.
Discrete Distribution Approximations Whenever a continuous distribution is used to approximate a discrete distribution, ½ interval corrections must be applied to the continuous distribution. Why?
½ Interval Corrections Table 5.2. Corrections for approximating a discrete random variable by a continuous random variable. DiscreteContinuous X = xX-1/2 < X < X+1/2 x < X < yX-1/2 < X < y+1/2 X < xX < x+1/2 X > xX > x-1/2 X < xX < X-1/2 X > xX > x+1/2
Binomial Distribution Additive property of the binomial If X is a binomial r.v. with parameters n 1 and p; and Y is a binomial r.v. with parameters n 2 and p, then Z = X + Y is a binomial r.v. with parameters n=n 1 +n 2 and p. Extending this to the sum of several binomial random variables, the C.L.T. indicates that the normal would approximate the binomial distribution if n is large.
Binomial Distribution Thus, as n get large: approaches a N(0,1).
Example X is a binomial r.v. with n= 50 and p= 0.2. Compare the normal approximation to the binomial fro evaluating prob(8
Negative Binomial Distribution Recall the additive property of the negative binomial: If X & Y are discrete r.v with f x (x;k 1,p) and f y (y;k 2,p) then for Z=X+Y, f z (z,k 1 +k 2,p). A negative binomial distribution with large k can be approximated by the normal distribution. approaches N(0,1) as k gets large.
Example Use both the negative binomial and the normal approximation to the negative binomial to find the probability that the 4 th occurrence of a 10 year flood will occur in the 40 th year?
Poisson Distribution Sum of 2 Poisson r.v. with parameters 1 and 2 is also a Poisson r.v. with parameter = Using C.L.T. For the sum of a large number of Poisson r.v. for large, Poisson may be approximated by which approaches N(0,1).
Continuous Distributions Many continuous distributions can be approximated by the normal distribution for certain value of their parameters. To make these approximations, the mean and variance of the distribution to be approximated are equated to the mean and variance of the normal. is N(0,1) if X is
Continuous Distributions Not all continuous distributions can be approximated by the normal. Things to look for are: 1. parameters that produce near zero skew 2. parameters that produce symmetry 3. parameters that produce distributions with tails that asymptotically approach p x (x) = 0 as X approaches large and small values.