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Physics 736 - Exercises #6
Physics 736 - Exercises #6
1 Instructions
The homework is due on Wednesday, April 7, 2010. Please leave
solutions in Karsten
Heeger's mailbox.
This homework will require the use of some plotting routine and/or
computing pack-
age such as Mathematica, ROOT, etc. Pick your favorite. Mathematica is installed
on the library computers.
Topics: probability density function, Monte Carlo, random numbers, covariance,
maximum likelihood.
2 Questions
1. Random Numbers { Generate random distributions of one thousand 2D and 3D
events. Make 2D and 3D scatter plots to illustrate the random
distribution of your
sequence of 1000 events. Describe what random number generator you or your soft-
ware package uses.
2. Illustrating the Central Limit Theorem { Show that whatever the
initial distribution,
a linear combination of a few variables almost always approximates to a Gaussian
distribution: Add together four random numbers ri , each distributed
uniformly and
independently in the range 0 to 1 to obtain a new variable z =
P
ri. Repeat the
procedure 50 times to obtain a set of 50 zj values. Plot these as a
histogram and
compare it with the appropriate Gaussian distribution. Give the mean
and variance
of this Gaussian distribution.
3. Simulating Radioactive Decay { This is a truly random process. The
probability of
decay is constant. The probability that a nucleus undergoes
radioactive decay in time
t is p, where p = t for t 1. The initial number of nuclei is
N0. Graph the
number of remaining nuclei as a function of time for the following cases:
N0 = 100, = 0:01 s1, t = 1 sec
N0 = 5000, = 0:03 s1, t = 1 sec
Show the results on both linear and logarithmic scales. Plot on the
same graphs the
expected decay curves.
4. Simulating Distributions { Write a program that generates the value
according
to the distribution function f() = (sin2 + acos2)1 in the range 0 2.
Generate 10000 values using a=0.5 and a=0.001. Plot the results for
each and overlay
the distribution curve, f(), properly normalized.
5. Correlation and Error Matrix { Generate 1000 events of uncorrelated
(x,y) values,
each given by a Gaussian distribution with x = 1, x = 2, y = 2, y
= 0:5. Show
a scatter plot of the data. Calculate the error matrix and the
correlation coecient,
, for this data set.
1
Consider the function f(x; y) = 5x + 8y. Evaluate the variance of this function
directly using the data set, and also by using the equation using the
error matrix.
Now rotate the events by 30 degrees around the center of the
distribution and repeat
above exercises.
6. Consider an experiment that measures the lifetime of an unstable
nucleus, N, using
the reaction A ! Ne, N ! Xp. The creation and decay of N is signaled by the
electron and proton. The lifetime of each N, which follows the
probability density
function (PDF) f = 1
et= , is measured from the time between observing the elec-
tron and proton with a resolution of t. The expected probability
density function
is the convolution of the exponential decay and the Gaussian resolution:
f(tj ; t) =
R 1
0
e
(tt0)2
22
t p
2t
et0=
dt0
Generate 200 events with = 1 and t = 0:5 sec. Use the maximum likelihood
method to nd ^ and the uncertainty, ^ . Plot the likelihood function and the
resulting PDF for the measured times compared to a histogram
containing the data.
Automate the maximum likelihood procedure so as to be able to repeat
this exercise
100 times, and plot the distribution of (^ )=^ for your 100
experiments and show
that it follows a unit Gaussian.
For one data sample assume that t is unknown and show a contour plot
in the ; t
plane with constant likelihood lnL = lnLmax 1=2.
2
1 Instructions
The homework is due on Wednesday, April 7, 2010. Please leave
solutions in Karsten
Heeger's mailbox.
This homework will require the use of some plotting routine and/or
computing pack-
age such as Mathematica, ROOT, etc. Pick your favorite. Mathematica is installed
on the library computers.
Topics: probability density function, Monte Carlo, random numbers, covariance,
maximum likelihood.
2 Questions
1. Random Numbers { Generate random distributions of one thousand 2D and 3D
events. Make 2D and 3D scatter plots to illustrate the random
distribution of your
sequence of 1000 events. Describe what random number generator you or your soft-
ware package uses.
2. Illustrating the Central Limit Theorem { Show that whatever the
initial distribution,
a linear combination of a few variables almost always approximates to a Gaussian
distribution: Add together four random numbers ri , each distributed
uniformly and
independently in the range 0 to 1 to obtain a new variable z =
P
ri. Repeat the
procedure 50 times to obtain a set of 50 zj values. Plot these as a
histogram and
compare it with the appropriate Gaussian distribution. Give the mean
and variance
of this Gaussian distribution.
3. Simulating Radioactive Decay { This is a truly random process. The
probability of
decay is constant. The probability that a nucleus undergoes
radioactive decay in time
t is p, where p = t for t 1. The initial number of nuclei is
N0. Graph the
number of remaining nuclei as a function of time for the following cases:
N0 = 100, = 0:01 s1, t = 1 sec
N0 = 5000, = 0:03 s1, t = 1 sec
Show the results on both linear and logarithmic scales. Plot on the
same graphs the
expected decay curves.
4. Simulating Distributions { Write a program that generates the value
according
to the distribution function f() = (sin2 + acos2)1 in the range 0 2.
Generate 10000 values using a=0.5 and a=0.001. Plot the results for
each and overlay
the distribution curve, f(), properly normalized.
5. Correlation and Error Matrix { Generate 1000 events of uncorrelated
(x,y) values,
each given by a Gaussian distribution with x = 1, x = 2, y = 2, y
= 0:5. Show
a scatter plot of the data. Calculate the error matrix and the
correlation coecient,
, for this data set.
1
Consider the function f(x; y) = 5x + 8y. Evaluate the variance of this function
directly using the data set, and also by using the equation using the
error matrix.
Now rotate the events by 30 degrees around the center of the
distribution and repeat
above exercises.
6. Consider an experiment that measures the lifetime of an unstable
nucleus, N, using
the reaction A ! Ne, N ! Xp. The creation and decay of N is signaled by the
electron and proton. The lifetime of each N, which follows the
probability density
function (PDF) f = 1
et= , is measured from the time between observing the elec-
tron and proton with a resolution of t. The expected probability
density function
is the convolution of the exponential decay and the Gaussian resolution:
f(tj ; t) =
R 1
0
e
(tt0)2
22
t p
2t
et0=
dt0
Generate 200 events with = 1 and t = 0:5 sec. Use the maximum likelihood
method to nd ^ and the uncertainty, ^ . Plot the likelihood function and the
resulting PDF for the measured times compared to a histogram
containing the data.
Automate the maximum likelihood procedure so as to be able to repeat
this exercise
100 times, and plot the distribution of (^ )=^ for your 100
experiments and show
that it follows a unit Gaussian.
For one data sample assume that t is unknown and show a contour plot
in the ; t
plane with constant likelihood lnL = lnLmax 1=2.
2
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