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