NORMAL OR GAUSSIAN DISTRIBUTION

Chapter 5

General Normal Distribution

Two parameter distribution with a pdf given by:

Xexpx

X ,2

1)(

2

2

1

2

1

22

Estimators for Q1 and Q2

Using MOM or Maximum Likelihood to estimate Q1 and Q2:

1 2

2

2

1

22

1)(

x

X exp

Properties of Normal Distribution

Bell-shapedContinuousSymmetrical about the mean

)0( s

Properties of the Normal Distribution

Mean and variance are sometimes referred to as location and scale parameters.

Varying s2 while holding m constant changes the scale but not the location of the distribution.

Varying m while holding s2 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, m and variance s2 as N(m,s2).

Reproductive Properties of the Normal Distribution

If X is a r.v., and , then the distribution of Y can be shown to be

If Xi for i = 1,2,…n are independently and normally distributed with mean mi and variance si

2 , then

is normally distributed with:

),( 2N bXaY ),( 22 bbaN

nnXbXbXbaY 2211

n

i iiY

n

i iiY

b

ba

1

222

1

Example

If xi is a random observation N(m,s2) what is the distribution of ?

n

i

i

n

xX

1

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).

xt

X dtexP

2

2

1

22

1)(

X

Z

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:

2

2

2

1)(

z

ezpZ

dtezPtZ

z2

2

2

1)(

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<z).

)();0();( zZzprobzZprobzZprob

Standard Normal Distribution

The table shows that 68.26% of normal distribution is within 1 s of the

mean. 95.44% of normal distribution is within 2 s of the

mean. 99.74% of normal distribution is within 3 s 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 3rd sigma bound. The probability of a value less than is only 0.0013.

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 m because the chance that X < 0 is many times (but not always) negligible.

3

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 Sn is the sum of n independently and identically distributed random variables Xi each having a mean, m, and variance, s2, then in the limit as n approaches infinity, the distribution of Sn approaches a normal distribution with mean nm and variance ns2 .

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.

Discrete Continuous X = x X-1/2 < X

< X+1/2 x < X < y X-1/2 < X

< y+1/2 X < x X < x+1/2 X > x X > x-1/2 X < x X < X-1/2 X > x X > x+1/2

Binomial Distribution

Additive property of the binomial If X is a binomial r.v. with parameters n1 and p;

and Y is a binomial r.v. with parameters n2 and p, then Z = X + Y is a binomial r.v. with parameters n=n1+n2 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:

)1( pnp

npXXZ

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<X<12).

Negative Binomial Distribution

Recall the additive property of the negative binomial: If X & Y are discrete r.v with fx(x;k1,p) and fy(y;k2,p) then for Z=X+Y, fz(z,k1+k2,p).

A negative binomial distribution with large k can be approximated by the normal distribution .

approaches N(0,1) as k

gets large.

2/ pkq

p

kX

XZ

Example

Use both the negative binomial and the normal approximation to the negative binomial to find the probability that the 4th occurrence of a 10 year flood will occur in the 40th year?

Poisson Distribution

Sum of 2 Poisson r.v. with parameters l1 and l2 is also a Poisson r.v. with parameter l = l1 + l2 .

Using C.L.T.For the sum of a large number of

Poisson r.v. for large l, Poisson may be approximated by

which approaches N(0,1).

XX

Z

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.

X

Z ),( 2Nis 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 skew2. parameters that produce symmetry3. parameters that produce distributions with tails

that asymptotically approach px(x) = 0 as X approaches large and small values.