Physics 310 The Method of Maximum Likelihood as applied to a linear function What is the appropriate condition to be satisfied for the sample population to best represent the parent population?

Physics 310 The parent population...  Suppose we have data which we comes from a parent population characterized by a linear relationship betwen x and y:  Suppose we have (x,y) data which we assume comes from a parent population characterized by a linear relationship betwen x and y:   y = a + bx  The problem: What is the best way to find the value of and which most likely represent the parent population from which our sample is obtained?  The problem: What is the best way to find the value of a and b which most likely represent the parent population from which our sample is obtained?

Physics 310 Why do we assume it is linear?  Suppose you have a table of data in two columns. What is the obvious next thing to do?  Suppose you have a table of (x,y) data in two columns. What is the obvious next thing to do?  Make a graph of vs and examine the apparant relationship between these pairs of measured values.  Make a graph of y vs x and examine the apparant relationship between these pairs of measured values.  Then, estimate the likely relationship -- i.e., “guess” the parent population!

Physics 310 Our goal...  If we assume the relationship between appears to be a linear one, our goal then is to find and  If we assume the relationship between (x,y) appears to be a linear one, our goal then is to find a and b  We will use the method of maximum likelihood -- as before, but now we determine two values, not just one.  The mathematical formalism is the same; the complexity is a bit greater!

Physics 310 Probability for one point -- Probability for one (x i,y i ) point --  The probability of making a measurement when there is an uncertainty in which isis just --  The probability of making a measurement (x i,y i ) when there is an uncertainty in y i which is  i is just --

Physics 310 The probability for all our points  The probability of getting our entire set of points is then --

Physics 310 To maximize the probability --  We must minimize to maximize the probability  We must minimize  2 to maximize the probability  This implies we minimize the squared deviations,  i.e., find the least squared deviations

Physics 310 Minimize with respect to...? Minimize  2 with respect to...?  In chaper 4, we minimized with respect to the trial mean,  In chaper 4, we minimized with respect to the trial mean,  ’  Here we minimize with respect to and simultaneously --  Here we minimize with respect to a and b simultaneously --  These lead to equations (6.10) and (6.11) - whose solutions are seen as (6.12)

Physics which give us and....which give us a and b.  Remember and are found such as to maximize the probability that these values of and will represent the parent population and from which our data came. This implies for each.  Remember a and b are found such as to maximize the probability that these values of a and b will represent the parent population a o and b o from which our data came. This implies for each x i. » Parent Population » Our data

Physics 310 Uncertainties and ? Uncertainties  a and  b ?  We estimate the uncertainties in and just as we did in chapter 4:  We estimate the uncertainties in a and b just as we did in chapter 4: – leading to equations (6.20) and (6.21).

Physics 310 Set up spreadsheet -- ,,,,,...  Then form columns of quotients  Sum the columns  Substitute in the equations to get the values of and, and and.  Substitute in the equations to get the values of a and b, and  a and  b.  Quote results as where  Quote results as y = a+ bx where a = MeV 0.04 MeV b = 23.6 MeV/ch 0.2 MeV/ch

Physics 310 We define as -- We define  2 as --  Calculate the individual for each point  Then sum to get the total  Then sum to get the total  2

Physics 310 What do we expect for ? What do we expect for  2 ?  Imagine that, on average, the deviations (numerator) were ~.  Imagine that, on average, the deviations (numerator) were ~ 1 .  Then, the sum of the ratio for datum points would be ~ !  Then, the sum of the ratio for N (x i,y i ) datum points would be ~ N!  Statistically, we expect for the value  Statistically, we expect for  2 the value

Physics 310 Conclusions...  What do you conclude if is ~1?  What do you conclude if is >>1?  What do you conclude if is <<1?  What do you do if one point has a very large value of ?  What do you do if one point has a very large value of  2 ?