Simple Probability Problem
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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?
xx
2 2xS
(true mean)
(true variance)
(sample mean)
(sample variance)
Sample vs Population
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.
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
-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
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
-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
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
Proper Choice of Δx
High K small Δx The choice of Δx is critical to the interpretation of the histogram.
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.
Figure 7.7
Frequency DistributionThe frequency distribution is a plot of nj /N versus magnitude. It is very similar to the histogram.
Histograms and Frequency Distributions in LabVIEW
‘digital’case
‘continuous’case