eee 25 lec 1
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EEE 25 Lecture 1TRANSCRIPT
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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EEE 25: Probability and Statistics for E&E Engineers Week 01 Lecture 01: Statistics & Data Analysis, Probability
Today
• Errors in Measurement
• Descriptive Statistics
• Frequency Distribution
• Stochastic Experiments
• Events
• Probabilities
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EEE 25: Probability and Statistics for E&E Engineers Week 01 Lecture 01: Statistics & Data Analysis, Probability
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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EEE 25: Probability and Statistics for E&E Engineers Week 01 Lecture 01: Statistics & Data Analysis, Probability
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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EEE 25: Probability and Statistics for E&E Engineers Week 01 Lecture 01: Statistics & Data Analysis, Probability
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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EEE 25: Probability and Statistics for E&E Engineers Week 01 Lecture 01: Statistics & Data Analysis, Probability
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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EEE 25: Probability and Statistics for E&E Engineers Week 01 Lecture 01: Statistics & Data Analysis, Probability
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
EEE 25: Probability and Statistics for E&E Engineers Week 01 Lecture 01: Statistics & Data Analysis, Probability
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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EEE 25: Probability and Statistics for E&E Engineers Week 01 Lecture 01: Statistics & Data Analysis, Probability
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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EEE 25: Probability and Statistics for E&E Engineers Week 01 Lecture 01: Statistics & Data Analysis, Probability
Measures of Central Tendency
• 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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EEE 25: Probability and Statistics for E&E Engineers Week 01 Lecture 01: Statistics & Data Analysis, Probability
Measures of Central Tendency
• 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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EEE 25: Probability and Statistics for E&E Engineers Week 01 Lecture 01: Statistics & Data Analysis, Probability
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 ?
EEE 25: Probability and Statistics for E&E Engineers Week 01 Lecture 01: Statistics & Data Analysis, Probability
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
EEE 25: Probability and Statistics for E&E Engineers Week 01 Lecture 01: Statistics & Data Analysis, Probability
Example
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Mean ? Mode ?
Median ?
EEE 25: Probability and Statistics for E&E Engineers Week 01 Lecture 01: Statistics & Data Analysis, Probability
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
EEE 25: Probability and Statistics for E&E Engineers Week 01 Lecture 01: Statistics & Data Analysis, Probability
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
EEE 25: Probability and Statistics for E&E Engineers Week 01 Lecture 01: Statistics & Data Analysis, Probability
Example
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Mean 6.08 Mode 7
Median 7
EEE 25: Probability and Statistics for E&E Engineers Week 01 Lecture 01: Statistics & Data Analysis, Probability
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
EEE 25: Probability and Statistics for E&E Engineers Week 01 Lecture 01: Statistics & Data Analysis, Probability
Measures of Variability
• The range of a sample is the difference between the highest and the lowest values of data – Sensitive to outliers
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EEE 25: Probability and Statistics for E&E Engineers Week 01 Lecture 01: Statistics & Data Analysis, Probability
Measures of Variability
• 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
EEE 25: Probability and Statistics for E&E Engineers Week 01 Lecture 01: Statistics & Data Analysis, Probability
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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EEE 25: Probability and Statistics for E&E Engineers Week 01 Lecture 01: Statistics & Data Analysis, Probability
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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EEE 25: Probability and Statistics for E&E Engineers Week 01 Lecture 01: Statistics & Data Analysis, Probability
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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EEE 25: Probability and Statistics for E&E Engineers Week 01 Lecture 01: Statistics & Data Analysis, Probability
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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EEE 25: Probability and Statistics for E&E Engineers Week 01 Lecture 01: Statistics & Data Analysis, Probability
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)
EEE 25: Probability and Statistics for E&E Engineers Week 01 Lecture 01: Statistics & Data Analysis, Probability
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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EEE 25: Probability and Statistics for E&E Engineers Week 01 Lecture 01: Statistics & Data Analysis, Probability
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
EEE 25: Probability and Statistics for E&E Engineers Week 01 Lecture 01: Statistics & Data Analysis, Probability
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
EEE 25: Probability and Statistics for E&E Engineers Week 01 Lecture 01: Statistics & Data Analysis, Probability
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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EEE 25: Probability and Statistics for E&E Engineers Week 01 Lecture 01: Statistics & Data Analysis, Probability
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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EEE 25: Probability and Statistics for E&E Engineers Week 01 Lecture 01: Statistics & Data Analysis, Probability
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( )+!
EEE 25: Probability and Statistics for E&E Engineers Week 01 Lecture 01: Statistics & Data Analysis, Probability
More Probability Properties
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P E( ) =1−P E( )
E
S
x
y
EEE 25: Probability and Statistics for E&E Engineers Week 01 Lecture 01: Statistics & Data Analysis, Probability
More Probability Properties
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P E∪F( ) = P E( )+P F( )−P E∩F( )
E
S
F
x y
z
w
EEE 25: Probability and Statistics for E&E Engineers Week 01 Lecture 01: Statistics & Data Analysis, Probability
More Probability Properties
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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
EEE 25: Probability and Statistics for E&E Engineers Week 01 Lecture 01: Statistics & Data Analysis, Probability
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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EEE 25: Probability and Statistics for E&E Engineers Week 01 Lecture 01: Statistics & Data Analysis, Probability
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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EEE 25: Probability and Statistics for E&E Engineers Week 01 Lecture 01: Statistics & Data Analysis, Probability
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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EEE 25: Probability and Statistics for E&E Engineers Week 01 Lecture 01: Statistics & Data Analysis, Probability
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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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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