1. Physics 114: Lecture 10 PDFs Part Deux Dale E. Gary NJIT Physics Department

2. February 18, 2010 Binomial & Poisson Distributions  The binomial distribution is  The mean is  The standard deviation is  The Poisson distribution is  The mean is  The standard deviation is  Writing only the coefficients, you begin to see a pattern: MatLAB: binopdf(x,n,p) MatLAB: poisspdf(x,  ) Use for yes/no statistics Use for counting statistics

3. February 18, 2010 Example 2.3  Some students measure some background counts of cosmic rays. They recorded numbers of counts in their detector for a series of 100 2-s intervals, and found a mean of 1.69 counts/interval. They can use the standard deviation formula from chapter 1, which is to get a standard deviation directly from the data. They do this and get s = 1.29. They can also estimate the standard deviation by  Now they change the length of time they count from 2-s intervals to 15-s intervals. Now the mean number of counts in each interval will increase. Now they measure a mean of 11.48, which implies while they again calculate s directly from their measurements to find s = 3.39.  We can plot the theoretical distributions using MatLAB poisspdf(x,mu), e.g. poisspdf(0:8,1.69) gives ans = 0.1845 0.3118 0.2635 0.1484 0.0627 0.0212 0.0060 0.0014 0.0003

4. February 18, 2010 Example 2.3, cont’d  The plots of the distributions is shown for these two cases in the plots at right.  You can see that for a small mean, the distribution is quite asymmetrical. As the mean increases, the distribution becomes somewhat more symmetrical (but is still not symmetrical at 11.48 counts/interval).  I have overplotted the mean and standard deviation. You can see that the mean does not coincide with the peak (the most probable value).    

5. February 18, 2010 Example 2.3, cont’d  Here is the higher-mean plot with the equivalent Gaussian (normal distribution) overlaid.  For large means (high counts), the Poisson distribution approaches the Gaussian distribution, which we will now describe further.

6. Gaussian or Normal Distribution  We will simply give the expression for the Gaussian distribution without derivation. Note that it is the limiting case of the Poisson distribution (counting statistics) as the mean  becomes large.  Unlike the binomial and Poisson distributions, which are defined only for integer values, the Gaussian distribution is continuous. That means it is a probability density function (pdf), and to get the probability that a value will fall between two values of x, you have to multiply by the bin width dx, or in the limit of infinitesimal bin widths, integrate:  Here, the mean  and standard deviation  are part of the definition of the distribution, so are not defined separately in terms of other parameters. February 18, 2010

7. Characteristics of the Gaussian  As always, the Gaussian pdf is normalized so that the area under the curve is unity (i.e. the integral from  to  is 1). The exponential itself has unit amplitude at x =  (i.e. exp(0) = 1 ), and if you do the integral of the exponential you will find that it is. Therefore, you have to divide by this factor to normalize the integral.  As you did in the first homework, you can determine the full width at half maximum for the Gaussian distribution by finding where the function falls to ½ its amplitude:  A good way to think about the standard deviation is that “most” values will lie within  of the mean. The actual percentage for  is 65%. If you go to , it is 95%. If you go to , it is 99.7%. We can think about the effect of this using a star image. February 18, 2010

8. Other Distributions  There are many other distributions that are met with in various circumstances, some phenomenological, and some based on theory. An interesting one is the Lorentzian distribution (or Cauchy distribution), which describes the shape of spectral lines in gases or plasmas such as the Sun.  This has a middle part that looks a little like a gaussian (the so-called line core), but the parts far from the mean (called the line wings) decrease slowly.  Although the mean is , the standard deviation of this distribution is undefined (i.e. the second moment is undefined) because of how slowly the distribution falls off (its integral is infinite). February 18, 2010