Physics 310

The Gaussian DistributionThe Gaussian Distribution

Physics 310

Section 2.2Section 2.2

Consider those (common) cases where the Consider those (common) cases where the probability of success in a probability of success in a singlesingle trial, trial, p, is , is notnot small, and, where small, and, where n is typically is typically largelarge??

Moreover, consider the situation where the Moreover, consider the situation where the number of successes number of successes x is is notnot restricted to be restricted to be an integer and an integer and x may be negativemay be negative..

What is the appropriate probability What is the appropriate probability distribution function to describe these cases?distribution function to describe these cases?

Physics 310

PG(x;) = ...

The Gaussian probability is:The Gaussian probability is:

PG(x;, ) =

12π

e− x−( )2

2 2

PG(x;,)

−∞

∞

∫ dx=1

−∞≤x≤∞

Physics 310

With unitless exponent...With unitless exponent...

One can define a unitless quatity One can define a unitless quatity z -- --

such that --such that --

x−( )

≡z

PG(z)=

12π

e−z2

2

PG(z)

−∞

∞

∫ dz=1

Physics 310

Some properties...Some properties...

The Gaussian probability distribution The Gaussian probability distribution function function – is a is a continuous functioncontinuous function in in x . .– is a is a normalizednormalized probability function. probability function.– is remarkably easy to use.is remarkably easy to use.– it is a differential probability function --it is a differential probability function --

dPG =PG(x;, )dx

Physics 310

Why use the Gaussian function?Why use the Gaussian function?

Why is it appropriate to use Why is it appropriate to use PG(x;)??

– One can show it is One can show it is consistent consistent with the Poisson with the Poisson probability function.probability function.

– Primarily, because Primarily, because it worksit works!!» That is, repeated measurements with random errors That is, repeated measurements with random errors

willwill be distributed according to a Gaussian be distributed according to a Gaussian probability. This is an probability. This is an empirical statementempirical statement, not a , not a theoretical one.theoretical one.

Physics 310

Gaussian curve overlaid on photon data

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Physics 310

Gaussian curve overlaid on photon data

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Series1

Series2dPG

dx

PGx

dPG = P(x;) dx

Physics 310

If the integrated area If the integrated area > 1......

If there are If there are N total entries in the plot, the total entries in the plot, the total integrated area under the data is total integrated area under the data is N..

We would want the total integrated area We would want the total integrated area under the Gaussian function also to be under the Gaussian function also to be N..

Or - Or - N PG(x;, )

−∞

∞

∫ dx=1• N

NPG(x;, )[ ]

−∞

∞

∫ dx=N

Physics 310

Now, for a histogram...Now, for a histogram...

In a histogram, you have In a histogram, you have binsbins over which the over which the variation in the probability is presumed to be ~ variation in the probability is presumed to be ~ constantconstant. The . The predicted numberpredicted number of events in each of events in each bin at bin at xi is -- is --

such that…such that…

Look at spreadsheet... NPG(xi ;, )[ ]

i=1

n

∑ Δx=N

NPG(xi;, )[ ] • Δx

Physics 310

Gaussian curve overlaid on photon data

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Series1

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Δx

NPGx

ΔPG = NP(x;) Δx

Physics 310

Central measures...Central measures...

Now, it is possible for us to determine a Now, it is possible for us to determine a mean and standard deviation for the mean and standard deviation for the Gaussian distribution. The following can Gaussian distribution. The following can be verified by direct integration:be verified by direct integration:

< x>= xPG(x;,)

−∞

∞

∫ dx=

< (x−)2 >= (x−)2 PG(x;, )

−∞

∞

∫ dx= 2

Physics 310

Full Width at Half-MaximumFull Width at Half-Maximum

A very useful and intuitive measure of the A very useful and intuitive measure of the Gaussian width is the FWHM, which is Gaussian width is the FWHM, which is described as the difference between the two described as the difference between the two symmetric values of x located on the symmetric values of x located on the Gaussian curve at values of the probability -Gaussian curve at values of the probability -

PG x±

Γ2

⎛ ⎝ ⎜

⎞ ⎠ ⎟= 0.5⋅PG(;,)[ ]

Physics 310

Gaussian curve overlaid on photon data

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Γ = 2.354FWHM

Max

0.5 Max

Physics 310

Integral Probability Distribution..Integral Probability Distribution..

PG(x;, )

−∞

x=a

∫ dx=Aa

Aa(x =a) ≤1

It is possible to obtain the It is possible to obtain the integral integral probabilityprobability for the Gaussian function as … for the Gaussian function as …

But, this integral must be done But, this integral must be done numericallynumerically..

Physics 310

Integral Probability Distribution..Integral Probability Distribution..

So, to find the integral probability for the So, to find the integral probability for the Gaussian function between arbitrary limits -Gaussian function between arbitrary limits -

one merely takes the difference --one merely takes the difference --

a≤x≤b

PG(x;, )

−∞

x=b

∫ dx− PG(x;, )−∞

x=a

∫ dx

= PG(x;, )

x=a

x=b

∫ dx

Physics 310

Integral Probability Distribution..Integral Probability Distribution..

So, how does one do this integration? So, how does one do this integration? – There are numerical routines which you can use There are numerical routines which you can use

to program it yourself, or, to program it yourself, or, – Use the MS Excel functionUse the MS Excel function

NORMDIST(x=a,F)

F = 0 -> value of PG(x=a; F = 1 -> value of Aa(x=a;

Physics 310

Integral Probability Distribution..Integral Probability Distribution..

So, one could choose So, one could choose a = , , b =

Or,Or,

PG(x;,)

−

+

∫ dx=A±1

PG(x;,)

−2

+2

∫ dx=A±2

Physics 310

Gaussian curve overlaid on photon data

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66.7%

95%

1

2

Physics 310

Integral Probability Distribution..Integral Probability Distribution..

In fact, you can integrateIn fact, you can integrate

for all values of for all values of x, and plot , and plot Aa vs vs x to get to get

the entire the entire IIntegral ntegral PProbability robability DDistributionistribution::

Look at spreadsheet...

PG(x;, )

−∞

x=a

∫ dx=Aa

Physics 310

Comments and observations...Comments and observations...

PG(x;) applies to applies to continuouscontinuous values of values of x..

It applies best to those cases for which It applies best to those cases for which n is is very large..

PG(x;) can be evaluated easily; it depends can be evaluated easily; it depends

only on only on and and .. It is a It is a normalizednormalized probability function. probability function.