Physics 257
Lecture 2:
Probability
Prof. Matt Dobbs
Physics Department, McGill U.
Prof. M. Dobbs,
Physics 257
Sept 20, 2007
2
Important Topics from Last Lecture
±
Covariance (from last week)
±
error propagation with partial derivatives
±
error on the mean
Δμ
=
σ
/
√
N
±
mean, weighted averages
1
)
)(
(
−
−
−
=
∑
N
y
y
x
x
s
i
i
i
xy
y
x
xy
xy
s
s
s
r
=
Prof. M. Dobbs,
Physics 257
Sept 20, 2007
3
Histograms
±
Area of bins is proportional to the number of occurrences.
±
divide full data range in n bins of width
Δ
x(bin±size±can
±also±
be variable)
z
choose bins carefully. In general
σ
/2 is a good choice.
¾
too narrow > few events in bin, fluctuation
¾
too wide > obscure details
Too narrow
Too wide
Just right.
Prof. M. Dobbs,
Physics 257
Sept 20, 2007
4
Assignment Feedback
the TA would “
be grateful if (the students)
would show their uniqueness everywhere
except in their handwriting
”
±
For every question (even computer responses), you
must show an example calculation. If you are using a
spreadsheet, you must print and include the
spreadsheet, labeling columns to show the formula
that has been used.
±
Late assignments (after 1:05pm on lecture day) will
lose 25%, like in the lab.
z
After lecture day the mark will be zero.
±
Remember,
σ
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View Full DocumentProf. M. Dobbs,
Physics 257
Sept 20, 2007
5
Outline
±
Questions ?
±
Why is s
α
1/(N1) ?
±
Probability distribution functions:
z
Flat distribution
z
Gaussian Distribution
z
Poisson Distrtribution
±
Famous Experimentalist
±
More fun with matlab
±
Probability language
Prof. M. Dobbs,
Physics 257
Sept 20, 2007
6
Review: std deviation, error on mean
±
Two researchers measure the period of a pendulum 9 times
each.
z
The researchers have thought about all possible systematic errors and
believe these are negligible.
±
Researcher1 measures:
t
1
, t
2
, …, t
9
z
mean = 10.0 seconds,
s
R1
= 0.6 seconds
z
The standard deviation s means: _____.
z
The error on the measured period is: _____________
±
Researcher2 measures:
t
1
, t
2
, …, t
9
z
mean = 10.2 seconds,
s
R1
= 0.6 seconds
z
The error on the measured period is: _____________
±
The two researchers combine their results and find an
improved measurement of the period: ____________
Prof. M. Dobbs,
Physics 257
Sept 20, 2007
7
Why s
α
1/(N1)?
±
Consider a Normal distribution with
μ
=1,
σ
=1
±
If we have just one sample of this distribution, x
1
=0.3, then we would
calculate <x>=0.3, and s=0 (the later is clearly wrong)
±
With 2 samples: 0.3,
1.5 we would calculate <x>=0.9 and
z
We are seeing a trend here – this equation would consistently
UNDERestimate the standard deviation, it is a BIASED estimator.
z
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 Fall '07
 Dobbs
 Physics, Normal Distribution, Prof. Matt Dobbs, Prof. M. Dobbs

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