eee 25 lec 1

EEE 25: Probability and Statistics for E&E Engineers Week 01 Lecture 01: Statistics & Data Analysis, Probability

Statistics and Data Analysis, Probability

Week 01 Lecture 01 EEE 25: Probability and Statistics

for Electrical and Electronics Engineers

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Today

•  Errors in Measurement

•  Descriptive Statistics

•  Frequency Distribution

•  Stochastic Experiments

•  Events

•  Probabilities

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Population vs. Sample

•  Population –  Collection of all possible data

•  Sample –  Subset of the population –  Characteristics of the sample will vary depending on the

values and number of samples taken

•  Example –  Battery lifetime in hours –  Individuals who received a BS in engineering in the

previous academic year

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Significant Figures

•  All the digits that are certain plus one digit which contains some uncertainty are said to be significant figures.

Examples: 1.00 0.25 0.05

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Rounding Off

•  Rule 1: If the remainder beyond the last digit to be reported is less than 5, drop the last digit.

•  Rule 2: If the remainder beyond the last digit is greater than 5, increase the last digit by 1.

•  Rule 3: If the remainder is exactly 5, round off to the closest even number.

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Kinds of Errors

•  Determinate –  can be attributed to definite causes –  methodic, operative or instrumental –  Example: weighing with uncalibrated weights

•  Indeterminate –  cannot be attributed to any known cause

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Accuracy and Precision

•  Accuracy is how closely a result agrees with the true result –  The smaller the error, the greater the accuracy

•  Precision refers to the agreement among a group of experimental results.

* Image from http://googleimage.xyz/accuracy-and-precision 7


Descriptive Statistics

•  Descriptive statistics –  the analysis of data that helps describe, show or summarize

data in a meaningful way –  a way of describing the characteristics of a large

collection of data by using a subset of it

•  Numerical summary values –  Measure of central tendency –  Measure of variability

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Measures of Central Tendency

•  The mean of a set of data is their arithmetic average.

•  Two flavors: –  Population mean, μ –  Sample mean

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sample in then observatioth - thesample in the nsobservatio ofnumber

mean sample

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•  The median of a sample is the value of the middle item when the items are arranged in increasing order.

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!x =x n+1( ) 2, if n is odd

12xn 2 + x n 2( )+1!"

#$, if n is even

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•  The mode is the most common value in the sample.

•  If all values occur equally frequently, then there is no mode.

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Example

20 20 80 40 10

60 60 100 80 70

10 10 70 40 90

50 100 70 60 10

10 60 10 50 70

80 50 80 50 60

70 20 90 60 70

60 10 80 60 60

20 20 30 40 70

50 70 100 10 60

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Mean ? Mode ?

Median ?


Example

10 20 50 60 80

10 20 50 60 80

10 20 60 70 80

10 30 60 70 80

10 40 60 70 80

10 40 60 70 90

10 40 60 70 90

10 50 60 70 100

20 50 60 70 100

20 50 60 70 100

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Mean 52.40 Mode 60

Median 60


Example

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Mean ? Mode ?

Median ?


Example

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VALUE FREQUENCY

1 2 2 7 3 4 4 4 5 1 6 7 7 9 8 7 9 4

10 7


Example

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VALUE FREQUENCY VALUE × FREQ LAST POSITION

1 2 2 2 2 7 14 9 3 4 12 13 4 4 16 17 5 1 5 18 6 7 42 25 7 9 63 34 8 7 56 41 9 4 36 45

10 7 70 52 52 316


Example

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Mean 6.08 Mode 7

Median 7


Measures of Variability

•  The measure of center reveals only partial information about a data set

•  Example: same mean/median, different spread

* Image taken from Devore 18



•  The range of a sample is the difference between the highest and the lowest values of data –  Sensitive to outliers

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•  Population variance, σ2

•  Sample variance, s2

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σ 2 =xi −µ( )2

i=1

N

∑N

s2 =xi − x( )2

i=1

n

∑n−1


Sample Spaces and Events

•  An experiment is any activity or process whose outcome is subject to uncertainty.

•  Examples –  Selecting a card from a deck –  Weighing a loaf of bread –  Obtaining blood types from a group of people

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Sample Spaces and Events

•  The sample space of an experiment is the set of all possible outcomes of that experiment. –  Discrete: elements are countable –  Continuous: elements are uncountable

•  An event is any collection (subset) of outcomes contained in the sample space.

•  We can use elementary set theory to study events.

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Set Definitions

•  a is an element of set A, a ∈ A

•  a is not an element of A, a ∈ A

•  B is a subset of A, B⊆A

•  B is a proper subset of A, B⊂A

•  ∅ is a null set

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Set Definitions

•  A = {x|0 ≤ x ≤ 1} –  uncountable and infinite

•  B = {0,1} –  countable and finite

•  N = {0,1,2,...} –  countably infinite

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Venn Diagram

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S is the universal set D is disjoint from the other sets B⊂A A ∩ C = shaded region (intersection)


Set Operations

•  If A ⊆B and B ⊆A, then A = B (equal)

•  A – B contains all elements in A that are not in B (difference)

•  ~A (or A) is the set of all elements not in A (complement)

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Set Operations

•  Commutative law

•  Distributive law

•  Associative law

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A∩B = B∩AA∪B = B∪A

A∪ B∩C( ) = A∪B( )∩ A∪C( )A∩ B∪C( ) = A∩B( )∪ A∩C( )

A∩ B∩C( ) = A∩B( )∩CA∪ B∪C( ) = A∪B( )∪C


DeMorgan’s Law

•  Duality Principle: If, in an identity, one replaces ∪ by ∩, ∩ by ∪, S by ∅, ∅ by S, and the sets by their complements, the identity is preserved.

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A∪B( ) = A∩B

A∩B( ) = A∪B


Events

Description Notation using sets An event is certain to occur S An event that is impossible ∅

Event A does not occur ~A Both events A and B occur A∩B Either A or B or both occurs A∪B If A occurs then B must also occur A⊆B

A and B are disjoint A∩B= ∅

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Probability

•  Given an experiment and a sample space S, the objective of probability is to assign to each event A a number P(A), called the probability of the event A, which will give a precise measure of the chance that A will occur.

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Axioms of Probability

•  AXIOM 1: For any event A , 1 ≥ P(A) ≥ 0 .

•  AXIOM 2: P(S) = 1.

•  AXIOM 3: If A1, A2, A3, … is an infinite collection of disjoint events, then

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P A1∪A2∪A3∪!( ) = P A1( )+P A2( )+P A3( )+!


More Probability Properties

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P E( ) =1−P E( )

E

S

x

y



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P E∪F( ) = P E( )+P F( )−P E∩F( )

E

S

F

x y

z

w



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P E1∪E2∪E3( ) = P E1( )+P E2( )+P E3( )−P E1∩E2( )−P E2∩E3( )−P E3∩E1( )+P E1∩E2∩E3( )

S E2

a b

c

d j

f

g h E3

E1


Example

•  A coin is tossed twice. –  What is the sample space?

–  What is the probability that at least 1 head occurs?

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Example

•  What is the probability of getting a total of 7 or 11 when a pair of fair dice is tossed? –  What is the sample space?

–  Let event A = total is 7. What is P(A)?

–  Let event B = total is 11. What is P(B)?

–  What is P(A or B)?

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Homework

•  A box contains three 9-V batteries and two 1.5-V batteries. A second box contains three 1-kΩ resistors and seven 10-kΩ resistors. The voltages and resistances are exact.

•  A battery from the first box and a resistor from the second box are picked at random. The two are connected to form a working circuit.

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Homework

•  What is the probability that the battery chosen is 9V?

•  What is the probability that the resistor chosen is 10kΩ?

•  What is the probability that the current is equal to 1.5 mA?

•  What is the probability that the current through the circuit is less than 1 mA?

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Statistics and Data Analysis, Probability

Week 01 Lecture 01 EEE 25: Probability and Statistics

for Electrical and Electronics Engineers

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eee 25 lec 1

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