Chapter 7: PROBABILITY
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Chapter 7: 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. Sir Francis Galton’s Quincunx
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Chapter 7: 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.” - PowerPoint PPT Presentation
Transcript of Chapter 7: PROBABILITY
Chapter 7: 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.
Sir Francis Galton’s Quincunx
Representing Population Behavior
Histogram
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, 3.0, 3.5, 4.0 V
Time record Histogram
N = 9 occurrences; j = 8 cells; nj = occurrences in j-th cell
The histogram is a plot of nj (ordinate) versus magnitude (abscissa).
Proper Choice of Δx
← made using 3histos.m
The choice of Δx is critical to the interpretation of the histogram.
Δx ChoicesTypically, we construct equal-width-interval histograms.
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.
Figure 7.7
Frequency DistributionThe frequency distribution is a plot of nj /N versus magnitude. It is very similar to the histogram.
← made using hf.m
Probability Density Function Concept
Figure 7.8
• Consider a signal that varies in time.
• What is the probability that the signal at a future time will reside between x and x + x?
Probability Density Function (pdf)
• Definition:
• For x(t):
• For n occurrences: fj /x
0 2 4 6 8 10 12 140
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
x
p(x)
Probability Density Function
In-Class Example• Determine the probability that x is between 1 and 7.
Probability Distribution Function (PDF)• The probability distribution function (PDF) is related to the integral of the pdf.
• A consequence of this is that
Figure 7.14
pdf PDF
Normalization of the pdf
1)( dxxpWhen the pdf is ‘normalized’ correctly:
Here,
>> not normalized
So, define pnew(x) = 1/3 p(x)
such that
In-Class Example• Determine the expressions for the PDF curve, knowing that of the pdf curve.
13713
71)(
721332
21
361
7212
21
361
xxx
xxxxP
• Integrating the pdf expressions give
0 2 4 6 8 10 12 140
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
x
p(x)
Probability Density Function
130
13713
711
10
)(361
361
x
xx
xx
x
xp
0 2 4 6 8 10 12 140
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
x
P(x
)
Probability Distribution Function
The Normal pdf and PDF
Figure 8.4
