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Lecture 3The Gaussian Probability Distribution

Function

- Introduction
- l The Gaussian probability distribution is

perhaps the most used distribution in all of

science. - Sometimes it is called the bell shaped curve

or normal distribution. - l Unlike the binomial and Poisson distribution,

the Gaussian is a continuous distribution - ?? mean of distribution (also at the same

place as mode and median) - ?2 variance of distribution
- y is a continuous variable (-8???y ??8?
- l Probability (P) of y being in the range a, b

is given by an integral - u The integral for arbitrary a and b cannot be

evaluated analytically. - The value of the integral has to be looked up

in a table (e.g. Appendixes A and B of Taylor).

Karl Friedrich Gauss 1777-1855

The integrals with limits -, can

be evaluated, see Barlow P. 37.

Plot of Gaussian pdf

p(x)

x

- l The total area under the curve is normalized

to one by the sv(2p) factor. - l We often talk about a measurement being a

certain number of standard deviations (?) away - from the mean (?) of the Gaussian.
- We can associate a probability for a

measurement to be ??- n? - from the mean just by calculating the area

outside of this region. - n? Prob. of exceeding mn?
- 0.67 0.5
- 1 0.32
- 2 0.05
- 3 0.003
- 4 0.00006
- Relationship between Gaussian and Binomial

distribution - l The Gaussian distribution can be derived from

the binomial (or Poisson) assuming - u p is finite
- u N is very large

It is very unlikely (lt 0.3) that a measurement

taken at random from a Gaussian pdf will be more

than 3s from the true mean of the distribution.

For a binomial distribution mean number of

heads ?? Np 5000 standard deviation ?

Np(1 - p)1/2 50 The probability to be

within m1??for this binomial distribution

is For a Gaussian distribution Both

distributions give about the same

probability! Why is the Gaussian pdf so

important? Þ Central Limit Theorem A crude

statement of the Central Limit Theorem

Things that are the result of the addition of

lots of small effects tend to become Gaussian. A

more exact statement Let Y1, Y2,...Yn be an

infinite sequence of independent random

variables each with the same probability

distribution. Suppose that the mean (?) and

variance (?2) of this distribution are both

finite. For any numbers a and b

The C.L.T. tells us that under a wide range

of circumstances the probability

distribution that describes the sum of random

variables tends towards a Gaussian

distribution as the number of terms in the sum

?8.

See Taylor10.4

Actually, the Ys can be from different pdfs!

How close to does n have to be??

- Alternatively we can write the CLT in a

different form - n ?m is sometimes called the error in the mean

(more on that later) - l For CLT to be valid
- The m and s of the pdf must be finite.
- No one term in sum should dominate the sum.
- l A random variable is not the same as a random

number. - A random variable is any rule that associates

a number with each outcome in S - (Devore, in probability and Statistics for

Engineering and the Sciences). - Here S is the set of possible outcomes.
- l Recall if y is described by a Gaussian pdf with

mean (m) of zero and s1 then the - probability that altyltb is given by

- l Example Generate a Gaussian distribution

using uniform random numbers. - u Random number generator gives numbers

distributed uniformly in the interval 0,1 - n m 1/2 and s2 1/12
- u Procedure
- Take 12 numbers (r1, r2,r12) from your

computers random number generator (ran(iseed)) - Add them together
- Subtract 6
- Þ Get a number that looks as if it is from a

Gaussian pdf!

A) 5000 random numbers

B) 5000 pairs (r1 r2) of random numbers

D) 5000 12-plets (r1 r2 r12) of random

numbers.

C) 5000 triplets (r1 r2 r3) of random numbers

E) 5000 12-plets (r1 r2 r12 - 6) of random

numbers.

E

Gaussian m 0 and s 1

Thus the sum of 12 uniform random numbers minus 6

is distributed as if it came from a Gaussian pdf

with m 0 and s 1.

0

-6

6

12 is close to !!

- l Example A watch makes an error of at most

1/2 minute per day. - After one year, whats the probability that the

watch is accurate to within 25 minutes? - u Assume that the daily errors are uniform in

-1/2, 1/2. - n For each day, the average error is zero and

the standard deviation 1/v12 minutes. - n The error over the course of a year is just

the addition of the daily error. - n Since the daily errors come from a uniform

distribution with a well defined mean and

variance - the Central Limit Theorem is applicable
- The upper limit corresponds to 25 minutes
- The lower limit corresponds to 25 minutes.
- The probability to be within 25 minutes is

This integral is 1 to about 3 part in 106!

- l Example The daily income of a card shark

has a uniform distribution in the interval

-40,50. - What is the probability that s/he wins more

than 500 in 60 days? - u Lets use the CLT to estimate this probability
- u The probability distribution of daily income

is uniform, p(y) 1. - p(y) needs to be normalized in computing the

average daily winning (?) and its standard

deviation (?). - u The lower limit of the winnings is 500

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