ET601!1!2 - Basic MATLAB Programming_2

7/23/2019 ET601!1!2 - Basic MATLAB Programming_2

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The word dice

52

Historically, dice is the plural of die.

In modern standard English, dice is used as both thesingular and the plural.

Example of 19th Century bone dice


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Advanced dice

53

[ http://gmdice.com/ ]
http://gmdice.com/http://gmdice.com/


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randi function

54

Generate uniformly distributed pseudorandom integers randi(imax) returns a scalar value between 1 andimax.

randi(imax,m,n) and randi(imax,[m,n])

return an m-by-n matrix containing pseudorandom integervalues drawn from the discrete uniform distribution on theinterval [1,imax].

randi(imax) is the same as randi(imax,1).

randi([imin,imax],...) returns an arraycontaining integer values drawn from the discrete uniformdistribution on the interval [imin,imax].

We have already seen the rand and randn functions.


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randi function: Example

55

close all; clear all;N = 1e3; % Number of trials (number of times that the coin is tossed)s = (rand(1,N) < 0.5); % Generate a sequence of N Coin Tosses.

% The results are saved in a row vector s.NH = cumsum(s); % Count the number of heads

plot(NH./(1:N)) % Plot the relative frequencies

LLN_cointoss.m

Same as

randi([0,1],1,N);


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Dice Simulator

56

http://www.dicesimulator.com/

Support up to 6 dice and also has some background

information on dice and random numbers.


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Two Dice

57


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Two-Dice Statistics

58


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Classical Probability

59

Assumptions The number of possible outcomes is finite.Equipossibility: The outcomes have equal probability of

occurrence. Equi-possible; equi-probable; euqally likely; fair

The bases for identifying equipossibility were often physical symmetry (e.g. a well-balanced die, made of homogeneous

material in a cubical shape)

a balance of information or knowledge concerning the various possibleoutcomes.

Formula: A probability is a fraction in which the bottomrepresents the number of possible outcomes, while the number ontop represents the number of outcomes in which the event ofinterest occurs.


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Two Dice

60

A pair of dice

Double six


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Two dice: Simulation

61

[ http://www2.whidbey.net/ohmsmath/webwork/javascript/dice2rol.htm ]
http://www2.whidbey.net/ohmsmath/webwork/javascript/dice2rol.htmhttp://www2.whidbey.net/ohmsmath/webwork/javascript/dice2rol.htm


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Two dice

62

Assume that the two dice are fair and independent.

P[sum of the two dice = 5] = 4/36


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Two dice

63

Assume that the two dice are fair and independent.


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Two-Dice Statistics

64


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Leibnizs Error (1768)

65

Though one of the finest minds of his age, Leibniz was notimmune to blunders: he thought it just as easy to throw 12with a pair of dice as to throw 11.

The truth is...

Leibniz is a German philosopher, mathematician,and statesman who developed differential andintegral calculus independently of Isaac Newton.

2[sum of the two dice = 11]36

1[sum of the two dice = 12]

36

P

P

[Gorroochurn, 2012]

0 100 200 300 400 500 600 700 800 900 10000

0.05

0.1

0.15

0.2

0.25


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Two dice: MATLAB Simulation

66

n = 1e3;Number_Dice = 2;D = randi([1,6],Number_Dice,n);S = sum(D); % sum along each column

RF11 = cumsum(S==11)./(1:n);plot(1:n,RF11)hold on

RF12 = cumsum(S==12)./(1:n);plot(1:n,RF12,'r')

0 100 200 300 400 500 600 700 800 900 10000

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

[SumofTwoDice.m]


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Concatenation

67

Concatenation is the process ofjoining arrays to make larger ones.

The pair of square brackets [] isthe concatenation operator.

Horizontal concatenation:

Concatenate arrays horizontallyusing commas.

Each array must have the samenumber of rows.

Vertical concatenation:

Concatenate arrays vertically usingsemicolons.

The arrays must have the samenumber of columns.


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More on plot: Line specification

68

Various line types (styles), plot symbols (markers) and colorsmay be obtained with plot(X,Y,S) where S is a character

string made from one element from any or all the following 3

columns:

y yellow . point - solid

m magenta o circle : dotted

c cyan x x-mark -. dashdot

r red + plus -- dashed

g green * star

b blue s square

w white d diamond

k black v triangle (down)

^ triangle (up)

< triangle (left)

> triangle (right)

p pentagram

h hexagram


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Character Strings

69

A character string is asequence of any number ofcharacters enclosed in singlequotes.

If the text includes a singlequote, use two single quoteswithin the definition.

These sequences are arrays,like all MATLAB variables.

Their class or data type ischar, which is short for

character. You can concatenate strings

with square brackets, just asyou concatenate numericarrays.

To convert numeric valuesto strings, use functions,

such as num2str or

int2str.


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More on plot : Examples

70

PLOT(X,Y,'c+:')plo

ts a cyan dotted line with a

plus at each data point

0 1 2 3 4 5 6 7 8 9 10-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

0 1 2 3 4 5 6 7 8 9 10-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

PLOT(X,Y,'bd')plots

blue diamond at each data point

but does not draw any line


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More on plot : Axes and Title

71

0 1 2 3 4 5 6 7 8 9 10-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1sin and cos functions

x

y

y = sin(x)

y = cos(x)

x = linspace(0,10,100);

y1 = sin(x);

y2 = cos(x);plot(x,y1,'mo--',x,y2,'bs--')

title('sin and cos functions')

xlabel('x')

ylabel('y')

legend('y = sin(x)','y = cos(x)')

grid on


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Two dice: MATLAB Simulation

72

[SumofTwoDice.m]

0 100 200 300 400 500 600 700 800 900 10000

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Number of Rolls

Relative

Frequency

Sum = 11Sum = 12

n = 1e3;Number_Dice = 2;D = randi([1,6],Number_Dice,n);S = sum(D); % sum along each column

RF11 = cumsum(S==11)./(1:n);plot(1:n,RF11)

hold onRF12 = cumsum(S==12)./(1:n);plot(1:n,RF12,'r')

xlabel('Number of Rolls')ylabel('Relative Frequency')legend('Sum = 11', 'Sum = 12')grid on

figureS_Support = (1*Number_Dice):(6*Number_Dice);hist(S,S_Support)

N_S_Sim = hist(S,S_Support);

2 3 4 5 6 7 8 9 10 11 120

20

40

60

80

100

120

140

160

180


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Loops

73

Loops are MATLAB constructs that permit us to execute asequence of statements more than once.

There are two basic forms of loop constructs:

while loops and

for loops. The major difference between these two types of loops is in

how the repetition is controlled.

The code in a while loop is repeated an indefinite number of

times until some user-specified condition is satisfied. By contrast, the code in a for loop is repeated a specified

number of times, and the number of repetitions is knownbefore the loops starts.


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for loop: a preview

74

Execute a block of statements a specified number of times.

k is the loop variable (also known as the loop index oriterator variable).

expr is the loop control expression, whose resultusually takes the form of a vector.

The elements in the vector are stored one at a time in the

variable k, and then the loop body is executed, so that the loopis executed once for each element in the array produced byexpr.

The statements between the for statement and the endstatement are known as the body of the loop. They areexecuted repeatedly during each pass of the for loop.

for = exprk bodyend

fork = 1:5x = k

end

for_ex1.m


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Two dice: Probability Calculation

75

Generate all the 36 possibilities.

Find the corresponding sum.

Count how many, among the

36, have a particular value.

Nested loops

Number_Dice = 2;Dice_Support = 1:6;S_Support = (1*Number_Dice):(6*Number_Dice);S = [];fork1 = Dice_Support

fork2 = Dice_SupportS = [S k1+k2];

endendSize_SampleSpace = length(S);

Number_11 = sum(S==11)Number_12 = sum(S==12)

% Count all possible cases at once

N_S = hist(S,S_Support)

P = sym(N_S)/Size_SampleSpace

Concatenation


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Vectorization and Preallocation

76

LLN_cointoss.m

LLN_cointoss_for_preallocated.m

close all; clear all;

N = 1e3; % Number of trials

s = zeros(1,N); % Preallocation

s(1) = randi([0,1]);

fork = 2:N

s(k) = randi([0,1]);

end

NH = cumsum(s); % Count the number of heads

plot(NH./(1:N)) % Plot relative frequencies

The script will still run without this line.

Revisiting an old script:

close all; clear all;N = 1e3; % Number of trials (number of times that the coin is tossed)s = randi([0,1],1,N); % Generate a sequence of N Coin Tosses.

% The results are saved in a row vector s.NH = cumsum(s); % Count the number of heads

plot(NH./(1:N)) % Plot the relative frequencies


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Preallocating Vectors (or Arrays)

77

Used when the content of a vector is computed/added/known while the loop isexecuted.

One method is to start with an empty vector and extend the vector by addingeach number to it as the numbers are computed. Inefficient.

Every time a vector is extended a new chunk of memory must be found

that is large enough for the new vector, and all of the values must be copied fromthe original location in memory to the new one. This can take a long time.

A better method is to preallocate the vector to the correct size and thenchange the value of each element to store the desired value. This method involves referring to each index in the output vector, and placing

each number into the next element in the output vector.

This method is far superior, if it is known ahead of time how many elements thevector will have.

One common method is to use the zeros function to preallocate the vector tothe correct length.


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zeros, ones, eye

78

zeros Create array of all zeros.

zeros returns the scalar 0. zeros(n) returns an n-by-n matrix of zeros. zeros(n,m) returns an n-by-m matrix of

zeros.

ones Create array of all ones.

eye Create identity matrix.

eye returns the scalar, 1.

eye(n) returns an n-by-n identity matrix withones on the main diagonal and zeros elsewhere.

eye(n,m) returns an n-by-m matrix with oneson the main diagonal and zeros elsewhere.


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Exercise

79

The following scriptwas used to calculatethe probabilities relatedto the sum resultingfrom a roll of two dice.

Modify the script hereso that the vector S ispreallocated.

Modify the script toalso create a matrix SS.

Its size is 6

6. Thevalue of its (k1,k2)element should bek1+k2.

Number_Dice = 2;Dice_Support = 1:6;S_Support = (1*Number_Dice):(6*Number_Dice);S = [];fork1 = Dice_Support

fork2 = Dice_SupportS = [S k1+k2];

endendSize_SampleSpace = length(S);

Number_11 = sum(S==11)Number_12 = sum(S==12)

% Count all possible cases at once

N_S = hist(S,S_Support)

P = sym(N_S)/Size_SampleSpace


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Galileo and the Duke of Tuscany

(1620)

80

When you toss three dice, the chance of the sum being 10 isgreater than the chance of the sum being 9.

The Grand Duke of Tuscany ordered Galileo to explain a paradox

arising in the experiment of tossing three dice:

Why, although there were an equal number of 6 partitions of thenumbers 9 and 10, did experience state that the chance of throwing a

total 9 with three fair dice was less than that of throwing a total of

10?

Partitions of sums 11, 12, 9 and 10 of the game of three fair dice:

[Gorroochurn, 2012]


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Exercise

81

Write a MATLAB script to Simulate N = 1000 repeated rolls of three dices.

Calculate the sum of the three dices from the rollsabove.

Plot the relative frequency for the event that thesum is 9.

In the same figure, plot (in red) the relativefrequency for the event that the sum is 10.

In another figure, create a histogram for the sumafter N rolls of three dice.

Calculate the actual probability for each possible

value of the sum. In another figure, compare the probability with the

relative frequency obtained after the N simulations.

2 4 6 8 10 12 14 16 180

0.02

0.04

0.06

0.08

0.1

0.12

0.14

sum of three dice

probability

relative frequency

0 100 200 300 400 500 600 700 800 900 10000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Number of Rolls

RelativeFrequen

cy

Sum = 11

Sum = 12


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Three-Dimensional Arrays

82

Arrays in MATLAB are not limited to two dimensions.


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Three-Dimensional Arrays

83

Three-dimensional arrays can be created directly using functionssuch as the zeros,ones, rand, and randi functions byspecifying three dimensions to begin with.

For example,zeros(4,3,2) will create a 432 matrix ofall 0s.


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Empty Array

84

An array that stores no value Created using empty square

brackets: []

Values can then be added by

concatenating. Can be used to delete elements

from vectors or matrices.

Individual elements cannot be

removed from matrices. Matrices always have to have the same

number of elements in every row.

Entire rows or columns could be

removed from a matrix.


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Scandal of Arithmetic

85

Which is more likely, obtaining at least one

six in 4 tosses of a fair dice (event A), or

obtaining at least one double six in 24 tosses

of a pair of dice (event B)?

44 4

4

2424 24

24

6 5 5( ) 1 .518

6 6

36 35 35( ) 1 .491

36 36

P A

P B

[http://www.youtube.com/watch?v=MrVD4q1m1Vo]


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Origin of Probability Theory

86

Probability theory was originally inspired by gamblingproblems.

In 1654, Chevalier de Mr invented a gambling systemwhich bet even money on case B.

When he began losing money, he asked his mathematicianfriend BlaisePascal to analyze his gambling system.

Pascal discovered that the Chevalier's system would loseabout 51 percent of the time.

Pascal became so interested in probability and together withanother famous mathematician, Pierre de Fermat, they laidthe foundation of probability theory.

best known for Fermat's Last Theorem


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Branching Statements if statement

87

General form:

The statements are executed if the real part of the expression has

all non-zero elements.

Zero or more ELSEIF parts can be used as well as nested ifs.

The expression usually contains ==, , =, or ~=.

if expressionstatements

elseif expression

statements

else

statementsend

The ELSE and ELSEIF parts

are optional.


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Example: Assigning Grades

88

Pass vs. Fail

% Generate a random number

score = randi(100, 1)

ifscore >= 50

grade = 'pass'else

grade = 'fail'

end

score 50true false

grade = pass grade = fail

if_ex_1.m


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Example: Assigning Letter Grades

89

score 80

score 70

score 60

score 50

true false

true false

true false

true false

grade = A

grade = B

grade = C

grade = D grade = F

% Generate a random numberscore = randi(100, 1)

ifscore >= 80

grade = 'A'

elseifscore >= 70

grade = 'B'

elseifscore >= 60

grade = 'C'

elseifscore >= 50

grade = 'D'

else

grade = 'F'

end


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Exercise

90

Write a MATLAB script to evaluate the relative frequenciesinvolved in the scandal of arithmetic.

0 100 200 300 400 500 600 700 800 900 10000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

number of rolls

relativefrequency

Event A: At least one six in 4 tosses of a fair dice

Event B: At least one double six in 24 tosses of a pair of dice


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Exercise

91

Write a MATLAB script to evaluate the relative frequenciesinvolved in the Monty Hall game.

0 100 200 300 400 500 600 700 800 900 10000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Number of Trials

RelativeFrequencyofWinning

Not Switch

Swicth

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ET601!1!2 - Basic MATLAB Programming_2