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%

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

NasNN

1

1

1

Note:

-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

50

20

x

xx

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

49.45

15.72

-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

49.86

21.46

Beware of small samples

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).

n5 = 3

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.

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.

Figure 7.7

fj = nj/N

n3

nj

f3

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

• odds to get something far from mean? • effect of noise form, e.g. uniform noise?