Download Probability and Statistics for Particle Physics

Probability and Statistics
for Particle Physics
Javier Magnin
CBPF – Brazilian Center for Research in
Physics
Rio de Janeiro - Brazil
Outline
• Course: three one hour lectures
• 1st lecture:
• General ideas / Preliminary concepts
• Probability and statistics
• Distributions
• 2nd lecture:
• Error matrix
• Combining errors / results
• Parameter fitting and hypothesis testing
• 3rd lecture:
• Parameter fitting and hypothesis testing (cont.)
• Examples of fitting procedures
st
1
lecture
Preliminary concepts
 Two types of experimental results:
• Determination of the numerical
value of some physical quantity
Parameter
determination
• Testing whether a particular
theory is consistent with data
Hypothesis
testing
 In real life there is a degree of overlapping between
both types above
 We will go through both types of results along these
lectures
Why estimate errors ?
Example:
• consider the accepted value of the speed of light
c = 2.998 x 108 m/s
• Assume that a new measurement gives
c´ = (3.09  x) x 108 m/s
Question:
• Are these two numbers consistent ?
Why estimate errors ?
Example:
• consider the accepted value of the speed of light
c = 2.998 x 108 m/s
• Assume that a new measurement gives
c´ = (3.09  x) x 108 m/s
• If x =  0.15  the new determination is consistent
with the accepted value.
• If x =  0.01  both values are inconsistent 
there is evidence for a change in the speed of light !
• If x =  2  both values are consistent, but
accuracy is so low that is impossible to detect a
change on c !
Random and systematic errors
Consider the experiment of determining the decay
constant  of a radioactive source:
 Count how many decays are observed in a time
interval t
 Determine the decay rate
Number of
decaying nuclei
• Random errors:
 inherent statistical error in counting events
 uncertainty in the mass of the sample
 timing of the period for which the decay are
observed
• Systematic errors:
 efficiency of the counter used to detect the
decays
 background (i.e. particles coming from other
sources)
 purity of the radioactive sample
 calibration errors
Probability
Suppose you repeat an experiment several times.
Even if you are able to keep constant the essential
conditions, the repetition will produce different
results
The result of an individual measurement will
be unpredictable even when the possible
results of a series of measurements have a
well-defined distribution
Definition:
The probability p of obtaining a specific result when
performing one measurement or trial is defined as
p=
Number of times on which that result occurs
Total number of measurements or trials
Rules of probability
1. If P(A) is the probability of a given event A, then
0  P(A)  1.
2. The probability P(A+B) that at least A or B occurs
is such that P(A+B)  P(A) + P(B). The equality is
valid only if A and B are exclusive events.
3. The probability P(AB) of obtaining both A and B is
P(AB) = P(A|B)P(B) = P(B|A)P(A), where P(A|B) is
the probability of obtaining A given that B has
occurred (P(A|B) is know as the conditional
probability of A given B).
4. The rule 3. defines the conditional probability as
P(A|B) = P(AB)/P(B).
Comments (about the rules !)
1. P(A+B) = P(A) + P(B) – P(AB)
to avoid double counting !
2. P(A|B) = P(AB)/P(B) =
(NC/N)/(NB/N) = NC/NB
3. P(A|B)  P(B|A)
4. If P(A|B) = P(A) then A and
B are independent, which is
equivalent to say that P(AB)
= P(A)P(B)
Example: use of conditional probability
Measurement of the mass difference m = m(KL) – m(KS)
from the K0p and K0p cross sections
p
p+
K0
p+
K0
p
K+
K0
+p
pK0
K+ + p  K0p+p
(production)
+p
K0  p+ + p(decay)
•K0 are detected in the
decay (event B)
•We want to measure
K0p  K0p (event A)
•We are interested in
P(AB) = P(B|A)P(A)
Bayes theorem
Let the sample space  be spanned by n mutually
exclusive and exhaustive sets Bi, i P(Bi) = 1. If
A is also a set belonging to , then
P(Bi|A) =
P(A|Bi) P(Bi)
j P(A|Bj)P(Bj)
Example (of the Bayes theorem)
Consider three drawers B1, B2, B3, each one with two
coins. B1 has two gold coins, B2 has one gold and one
silver and B3 has two silver coins.
 Now select a random drawer and pick a coin from it.
 Supposing that the coin is gold, what is the probability
of having a second gold coin in the same drawer (or, what
is the probability of having selected the drawer B1, given
that I selected a gold coin)?
B1
B2
B3
If A is the event of first picking a gold coin, then
P(A|B1) = 1; P(A|B2) = 1/2; P(A|B3) = 0
Since the drawers is selected at random,
P(B1) = P(B2) = P(B3) = 1/3
Hence, from Bayes theorem follows that
P(B1|A) =
P(A|B1) P(B1)
j P(A|Bj)P(Bj)
=
1 x 1/3
1x1/3 + 1/2x1/3 + 0x1/3
= 2/3
Thus, although the probability of selecting the drawer B1
is 1/3, the observation that the first coin drawn was gold
doubles the probability that the drawer B1 had been
selected
Statistics
Probability: we start with a well defined problem,
and calculate from this the possible outcome of a
specific experiment
Statistics: we use data from a experiment to
deduce what are the rules or laws relevant to the
experiment
Statistics
Statistics  two different problems (with some overlap)
Parameter determination: use data to measure the
parameter of a given model or theory
Hypothesis testing: use data to test the hypothesis
of a model
Probability
Statistics
From theory to data
From data to theory
•Toss a coin. If P(heads)
= 0.5, how many heads
you get in 50 tosses?
•Given a sample of K*
mesons of known
polarization, what is the
forward/backward
asymmetry?
•If you observe 27 heads
in 50 tosses, what is the
value of P(heads)? (*)
•Of 1000 K* mesons, 600
were observed that decay
forward. What is the K*
polarization? (**)
(*) Parameter determination: deduce a quantity and its error
(**) Hypothesis testing: check a theory
Distributions
In general, the result of repeating
the same measurement many times
does not lead to the same result
Experiment:
• measure the length of one side of your table
10 times and display the results in a
histogram.
• What happens if you repeat the
measurement 50 times ?
Distribution n(x): describes
how often a value of the
variable x occurs in a sample
Measuring the table...
Sample: the total number of
measurements of the side
of the table, N
n
x
n
Bin size
x
x  discrete or
continuous variable
distributed in a finite
or infinite range
n
x
Mean and variance: center and
width of a distribution
Assume a set of N separate measurements of the same
physical quantity
Mean
Variance
Mean and variance: center and
width of a distribution
Assume a set of N separate measurements of the same
physical quantity
Mean
Variance
Convention:
m and s2 are the true
values of the mean
and the variance resp.
Entries in
each x bin
(66 in total)
The mean x is known to
an accuracy of s/ N
(more about 1/ N later)
Then s2 will not change
by increasing N, but the
variance of the mean,
s2/ N, decreases with
N
s2 is a measurement of how
the distribution spreads out
and not a measurement of
the error in x !
x is better determined as N increases
s2 is a property of the distribution
Continuous distributions
Number of events
Mean
Variance
N large
Special distributions
Binomial distribution
N independent trials, each of which has
only two possible outcomes, success p or
failure (1-p)
Probability of failure in
the remaining N-r trials
Ordering of the
successes and failures
Probability of obtaining
successes on r attempts
Symmetric if
p=(1-p)
r = non negative integer
0 < p < 1, p Real
Mean:
Variance:
Where the binomial distribution apply ?
 Throw a dice N times. What is the probability of
obtaining a 6 on exactly r occasions ?
 The angles that the decay products from a given
source make with some fixed axis are measured. If
the expected distribution is known, what is the
probability of observing r decays in the forward
hemisphere ( < p/2) from a total sample of N decays ?
Properties and limiting cases


 Poisson

 Gaussian
Poisson distribution
Probability of observing r independent events in a time
interval t, when the counting rate is m and the expected
number of events in the time interval is 
Limiting case of the
binomial distribution
when
r non negative integer
 positive real number
Mean = Variance = 
Where the Poisson distribution apply ?
 Particle emission from a radioactive source: if
particles are emitted from the source at an average
rate of  (= number of particles emitted by unit time),
the number of particles emitted, r, in a time interval
t follows a Poisson distribution.
Additive property of independent Poisson variables
Assume that you have a radioactive emitting source in a
medium where there exist radioactive background
emissions at a rate mb. The radioactive source emits at a
ratio mx
Radioactive source
Background
What is the distribution probability for the emission
of source + background ?
r = number of particles emitted by the source + background
Binomial formula
Gaussian distribution
General form:
Continuous variable
Mean
Variance
Normalization
Standard form
Properties
I[m-s,m+s] ~ 0.68 I[-,+ ]
• Symmetric w.r.t. m
• if xi is gaussian with m
and s2 then
is gaussian with m and
s2/n
Where the Gaussian distribution apply ?
 The result of repeating an experiment many
times produces a spread of answers whose
distribution is approximately Gaussian.
 If the individual errors contributing to the final
answer are small, the approximation to a Gaussian
is especially good.
N  ; Np = m
Binomial
Poisson
N
m
Gaussian