Chapter 7 - Central Limit Theorem - University of Kentucky

[Pages:10]Topics

Part 1 ? Single measurement

1.

Basic stuff (Chapter 1 and 2)

2.

Propagation of uncertainties (Chapter 3)

Part 2 ? Multiple measurements as independent results

1.

Mean and standard deviation (Chapter 4)

2.

Basic on probability distribution function (not in text explicitly)

3.

The Binomial distribution (Chapter 10)

4.

The Poisson distribution (Chapter 11)

5.

Normal distribution (first half of Chapter 5)

6.

2 test ? how well does the data fit the distribution model? (Chapter 12)

Part 3 ? Multiple measurements as one sample

1.

Central limit theorem (not in text explicitly)

2.

Normal distribution (second half of Chapter 5)

3.

Propagation of error (Chapter 3)

4.

Rejection of data (Chapter 6)

5.

Merging two sets of data together (Chapter 7)

Part 4 - Dependent variables

1.

Curve fitting (Chapter 8)

2.

Covariance and correlation (Chapter 9)

One Single Experiment of n Measurements

In one single experiment, you measure a random variable (x) N times and obtain n results, x1, x2, ......., xN. We know how to: (i) Calculate the mean of these N readings (but do not know why). (ii) Calculate the sample standard deviation of these N readings (but do not know why). (iii) Calculate the population standard deviation to see how spread is these N readings. (iv) Make a histogram and expect it will follow some kind of probability density function.

In here, we just focus on calculating the mean of these N readings. i.e. the mean of these N readings will itself become a random variable if we repeat the experiment.

Many Experiments of n Measurements for Each Experiment

Now we perform the same experiment as in the last slide for n times. In

each experiment, we measure a random variable (x) N times and obtain

n results, x1, x2, ......., xN. We then calculate the mean x of these N readings.

So after N experiments, we will have performed N2 measurements and have N means:

Experiment 1: x11, x21, ......., xN1 Experiment 2: x12, x22, ......., xN2 :

mean x1 mean x2

Experiment N: x1N, x2N, ......., xNN mean xN

New random variable

Central Limit Theorem (Unified Theory of Data Analysis)

For the N readings in each experiment x1, x2, ......., xN they may follow some kind of distribution function, but the mean x of these N experiment will always follow the Normal distribution GX,(x) if N is large (ideally N). Note that the mean of each experiment is now the random variable of this Normal distribution.

Experiment 1: x11, x21, ......., xN1 Experiment 2: x12, x22, ......., xN2 :

mean x1 mean x2

Experiment N: x1N, x2N, ......., xNN mean xN

Follow some kind of distribution function

Follow Normal distribution G X,(x)

But what are the values of X and ?

If you know the distribution function of x1, x2, ......., xN (unlikely)

But what are the values of X and in the Normal

distribution G X,(x)?

Suppose we know the distribution function of x1, x2, ......., xN, say, P(x). This distribution has its own mean x and

variance x2 : x = xP(x) dx

2 x

=

(x - x )2 P(x) dx

Central limit theorem state that:

X = x

2

=

2 x

N

Note: divided by N.

Unfortunately we rarely know the distribution function P(x).

If you don't know the distribution function of x1, x2, ......., xN (likely)

We can guess! Suppose we have x1, x2, ......., xN in one experiment:

Best Estimate of X = x = x1 + x2 +Lx N N

(mean of the set of data)

(Eq. 5.42 of text)

Best Estimate of x (or N ) =

(x1 - x)2 + (x2 - x)2 + L(x N - x)2 (N -1)

(sample standard deviation of the set of data)

(Eq. 5.45 of text)

Note what we are estimating: parameters X and of the normal distribution.

Best Estimate of X (slide 1)

Suppose we have x1, x2, ......., xn in one experiment. For simplicity, we assume (not necessary) the distribution

function of these data is P(x) Gx ,x (x). According to the Central Limit Theorem, the mean of these data point with follow (necessary) Normal distribution function GX , (x).

The theorem also requires x = X and x/ N = .

Pr ob(x1) = x

1

2

e-

(x1 -x)2 2 x 2

dx1

Pr ob(x2 ) = x

1

2

e-

(x2 -x)2 2 x 2

dx 2

M

Pr ob(x N ) = x

1

2

e-

(x 2 -x)2 2 x 2

dx N

Best Estimate of X (slide 2)

Pr ob (x1, x2 ,L, x N ) = Pr ob(x1) Pr ob(xN )LPr ob(xN )

=

x

1

2

e-

(x1 -x) 2 x 2

2

dx 1

x

1

2

e-

(x2 -x) 2 x 2

2

dx

2

L

x

1

2

e-

(x 2 -x)2 2 x 2

dx

N

( ) =

1

N x

2

e dx dx Ldx -

(x1

-

x)2

+

(x

2 -x)2 2 x 2

+L(x

N

-

x)

2

N

12

N

2

Principle of maximum likelihood :

x

Pr

ob

(x 1

,

x

2

, L,

x

N

)

=

0

2(x 1

-

x)

+

2(x2 - x) +L+

2 x 2

2(x N

-

x)

=

0

(x1 - x) + (x2 - x) +L+ (x N - x)

Nx = x1 + x2 +L + x N

x = x1 + x2 +L+ xN N

Best estimate of X = x = x1 + x2 +L+ x N N

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