Physics 310 The Gaussian Distribution

Physics 310 Section 2.2 Consider those (common) cases where the probability of success in a single trial,, is not small, and, where is typically large? Consider those (common) cases where the probability of success in a single trial, p, is not small, and, where n is typically large? Moreover, consider the situation where the number of successes is not restricted to be an integer and may be negative. Moreover, consider the situation where the number of successes x is not restricted to be an integer and x may be negative. What is the appropriate probability distribution function to describe these cases? What is the appropriate probability distribution function to describe these cases?

Physics 310 P G (x;  ) =... The Gaussian probability is: The Gaussian probability is:

Physics 310 With unitless exponent... One can define a unitless quatity -- One can define a unitless quatity z -- such that -- such that --

Physics 310 Some properties... The Gaussian probability distribution function The Gaussian probability distribution function –is a continuous function in. –is a continuous function in x. –is a normalized probability function. –is remarkably easy to use. –it is a differential probability function --

Physics 310 Why use the Gaussian function? Why is it appropriate to use ? Why is it appropriate to use P G (x;  )? –One can show it is consistent with the Poisson probability function. –Primarily, because it works! »That is, repeated measurements with random errors will be distributed according to a Gaussian probability. This is an empirical statement, not a theoretical one.

Physics 310

dP G dx P G  x  dP G = P(x;  ) dx

Physics 310 If the integrated area... If the integrated area > 1... If there are total entries in the plot, the total integrated area under the data is. If there are N total entries in the plot, the total integrated area under the data is N. We would want the total integrated area under the Gaussian function also to be. We would want the total integrated area under the Gaussian function also to be N. Or - Or -

Physics 310 Now, for a histogram... In a histogram, you have bins over which the variation in the probability is presumed to be ~ constant. The predicted number of events in each bin at is -- In a histogram, you have bins over which the variation in the probability is presumed to be ~ constant. The predicted number of events in each bin at x i is -- such that… such that… Look at spreadsheet...

Physics 310 PGPG xx NP G  x   P G = NP(x;  )  x

Physics 310 Central measures... Now, it is possible for us to determine a mean and standard deviation for the Gaussian distribution. The following can be verified by direct integration: Now, it is possible for us to determine a mean and standard deviation for the Gaussian distribution. The following can be verified by direct integration:

Physics 310 Full Width at Half-Maximum A very useful and intuitive measure of the Gaussian width is the FWHM, which is described as the difference between the two symmetric values of x located on the Gaussian curve at values of the probability - A very useful and intuitive measure of the Gaussian width is the FWHM, which is described as the difference between the two symmetric values of x located on the Gaussian curve at values of the probability -

Physics 310  =  FWHM Max 0.5 Max

Physics 310 Integral Probability Distribution.. It is possible to obtain the integral probability for the Gaussian function as … It is possible to obtain the integral probability for the Gaussian function as … But, this integral must be done numerically. But, this integral must be done numerically.

Physics 310 Integral Probability Distribution.. So, to find the integral probability for the Gaussian function between arbitrary limits - So, to find the integral probability for the Gaussian function between arbitrary limits - one merely takes the difference -- one merely takes the difference --

Physics 310 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 to program it yourself, or, –Use the MS Excel function NORMDIST(x=a,  F) F = 0 -> value of P G (x=a;   F = 1 -> value of A a (x=a; 

Physics 310 Integral Probability Distribution.. So, one could choose, So, one could choose a = , b =  Or, Or,

Physics % 95% 11 22

Physics 310 Integral Probability Distribution.. In fact, you can integrate In fact, you can integrate for all values of, and plot vs to get the entire Integral Probability Distribution: for all values of x, and plot A a vs x to get the entire Integral Probability Distribution: Look at spreadsheet...

Physics 310 Comments and observations... Comments and observations... applies to continuous values of. P G (x;  ) applies to continuous values of x. It applies best to those cases for which is. It applies best to those cases for which n is very large. can be evaluated easily; it depends only on and. P G (x;  ) can be evaluated easily; it depends only on  and . It is a normalized probability function. It is a normalized probability function.