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Probability
“When you deal in large numbers, probabilities are the same
as certainties. I wouldn’t bet my life on the toss of a single
coin, but I would, with great confidence, bet on heads
appearing between 49 % and 51 % of the throws of a coin if
the number of tosses was 1 billion.”
Brian Silver, 1998, The Ascent of Science, Oxford University
Press.
Simple Probability Problem
• Imagine I randomly choose 2 people from this
class. What is the probability that both are in the
same laboratory section?
• Assume: 99 students, all present; 9 lab sections,
all equally populated  11 students per lab
section
• Choose 1st student (note this choice can’t be
wrong)
• Now there are 98 students left and 10 that are in
the same section as the first…
• Thus the answer is 10/98 = 10.2%
Sample vs Population
x

2
(true mean)
(true variance)
(sample mean)
x
(sample variance)
2
Sx
Populations Parameters and
Sample Statistics
• Population parameters include its true mean, variance
and standard deviation (square root of the variance):
x  lim N 
1
N
 2  lim N 
N
x
1
N
i 1
i
N
2
(
x

x
)
 i
i 1
• Sample statistics with statistical inference can be used
to estimate their corresponding population parameters
to within an uncertainty.
Populations Parameters and
Sample Statistics
• A sample is a finite-member representation of an
‘infinite’-member population.
• Sample statistics include its sample mean, variance
and standard deviation (square root of the variance):
1
x
N
N
x
i 1
i
N
1
2
S x2 
(
x

x
)

i
N  1 i 1
Note:
1
1

as N  
N 1 N
Normally Distributed Population
using MATLAB’s command randtool
Distribution
Samples
4500
4000
3500
Counts
x  
x  
3000
x
x 
50

20
2500
2000
1500
1000
500
0
-100
-50
0
50
Values
100
150
200
Random Sample of 50
Distribution
Samples
18
16
x  49.45
S x  15.72
14
Counts
12
10
8
6
4
2
0
-100
-50
0
50
Values
100
150
200
Another Random Sample of 50
Distribution
Samples
25
x  49.86
S x  21.46
20
Counts
15
x
x
Sx 
10
5
0
-100
-50
0
50
Values
100
150
200
Beware of small samples
The Histogram
Figure 7.3
Time record
Figure 7.4
Histogram of digital data
analog,
discrete, and
digital signals
10 digital values: 1.5, 1.0, 2.5, 4.0, 3.5, 2.0, 2.5, 3.0, 2.5 and 0.5 V
resorted in order: 0.5, 1.0, 1.5, 2.0, 2.5, 2.5, 2.5, 3.0, 3.5, 4.0 V
N = 9 occurrences; j = 8 cells; nj = occurrences in j-th cell
n5 = 3
The histogram is a plot of nj (ordinate) versus magnitude (abscissa).
Proper Choice of Δx
High K  small Δx
The choice of Δx is critical to the interpretation of the histogram.
theoretical values
data (5000
randomly drawn
values)
Figure 7.5
Histogram Construction Rules
To construct equal-width histograms:
1. Identify the minimum and maximum values of x and its range
where xrange = xmax – xmin.
2. Determine K class intervals (usually use K = 1.15N1/3).
3. Calculate Δx = xrange / K.
4. Determine nj (j = 1 to K) in each Δx interval. Note ∑nj = N.
5. Check that nj > 5 AND Δx ≥ Ux.
6. Plot nj versus xmj,where xmj is the midpoint value of each interval.
Frequency Distribution
The frequency distribution is a plot of nj /N versus magnitude.
It is very similar to the histogram.
n3
nj
f3
fj = nj/N
Figure 7.7
Histograms and Frequency Distributions in LabVIEW
‘digital’
case
‘continuous’
case
• odds to get something far from mean?
• effect of noise form, e.g. uniform noise?