simple probability problem
DESCRIPTION
Simple Probability Problem. Imagine I randomly choose 2 people from this class. What is the probability that both are in the same laboratory section?. (true mean). (sample mean). (sample variance). (true variance). Sample vs Population. Populations Parameters and Sample Statistics. - PowerPoint PPT PresentationTRANSCRIPT
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Simple Probability Problem
• Imagine I randomly choose 2 people from this class. What is the probability that both are in the same laboratory section?
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xx
2 2xS
(true mean)
(true variance)
(sample mean)
(sample variance)
Sample vs Population
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Populations Parameters and Sample Statistics
• Population parameters include its true mean, variance and standard deviation (square root of the variance):
2
1
2
1
)(1
lim
1lim
xxN
xN
x
N
iiN
N
iiN
• Sample statistics with statistical inference can be used to estimate their corresponding population parameters to within an uncertainty.
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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):
2
1
2
1
)(1
1
1
xxN
S
xN
x
N
iix
N
ii
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-100 -50 0 50 100 150 2000
500
1000
1500
2000
2500
3000
3500
4000
4500
Cou
nts
Values
SamplesDistribution
x
Normally Distributed Populationusing MATLAB’s command randtool
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xS
x
-100 -50 0 50 100 150 2000
2
4
6
8
10
12
14
16
18
Cou
nts
Values
SamplesDistribution
Random Sample of 50
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-100 -50 0 50 100 150 2000
5
10
15
20
25
Cou
nts
Values
SamplesDistribution
xx
xS
x
xS
x
Another Random Sample of 50
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The 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, 2.5, 3.0, 3.5, 4.0 V
Time record Histogram of digital data
N = 9 occurrences; j = 8 cells; nj = occurrences in j-th cell
The histogram is a plot of nj (ordinate) versus magnitude (abscissa).
Figure 7.3 Figure 7.4
analog,discrete, and digital signals
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Proper Choice of Δx
High K small Δx The choice of Δx is critical to the interpretation of the histogram.
Figure 7.5
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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.
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Figure 7.7
Frequency DistributionThe frequency distribution is a plot of nj /N versus magnitude. It is very similar to the histogram.
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Histograms and Frequency Distributions in LabVIEW
‘digital’case
‘continuous’case