Normal Distribution [Miller 1985 n Martinez 2001]

History:

1. Known as Gaussian Distribution.
2. Studied first in 18th century when scientists observed an astonishing degree of regularity in errors of measurement.
3. The error distributions observed were approximated by distribution called ' normal curve of errors' (Bell shape) produced by the normal distrbution Eqn. that determined by the expected value and variance for normal distribution.

Properties:

1. PDF aproaches zero as x approaches + or - inf
2. centered at mean μ and max value occur at x=μ
3. PDF for normal distribution is symmetric about mean μ

MATLAB Command:

1. normcdf(x,mu,sigma)
2. normpdf(x.mu,sigma)
3. normspec(specs, mu, sigma)

MATLAB Example:
%Set up parameter for normal distb.
mu = 5;
sigma = 2;
%Set up upper and lower limit specs
specs = [2, 8]prob = normspec(specs, mu, sigma);

Equations:

GAUSSIAN ( NORMAL ) DISTRIBUTION ( PDF ), MEAN, AND VARIANCE


STANDARDIZED RAN VAR, STANDARD NORMAL DISTRIBUTION ( CDF )

Definition: Standard Nornal Distribution is a normal probability distribution that has a mean of 0 and a standard deviation of 1

GAUSSIAN APPROXIMATION TO THE BINOMIAL DISTRIBUTION

Use to approximate the binomial distribution when n is large but but is close to 0.5, not small enough to use Poisson Approximation.
Rule of thumb: use the normal approximation to the binomial distribution only when np and (1-np) are both greater than 5.

Theorem: (State without proof) If x is a value of random variable having the binomial distribution with the parameters n and p and if
then the limiting form of the distribution function of this standardized random variables as n --> inf is given by