Lecture # 8
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GENG 200
Probability and Statistics
Lecture Note #8
Dr. Adnan Abu-Dayya
Electrical Engineering Department
Qatar University
Chapter 2:
Probability
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Chapter 2 Outline
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1.
Sample Space and Events
2.
Counting Technique
3.
Interpretations and Axioms of Probability
4.
Union of Events and Addition rules
5.
Conditional probability
6.
Intersection, Multiplication and Total probability rules
7.
Independence
8.
Bayes’ theorem
9.
Random variables
Random Variable and its Notation
▪ A variable that associates a number with the outcome of a random
experiment is called a random variable
▪ A random variable is a function that assigns a real number to each
outcome in the sample space of a random experiment
▪ A random variable is denoted by an uppercase letter such as X. After the
experiment is conducted, the measured value of the random variable is
denoted by a lowercase letter such as
x = 70 milliamperes. e.g., P(X = x)
SEC 2-8 RANDOM VARIABLES
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Example
▪ A batch of 500 machined parts contains 10 that do not conform to
customer requirements. The random variable is the number of parts
in a sample of 15 parts that do not conform to customer
requirements. Determine the range (possible values) of the random
variable
Answer:
The range of X is {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
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Discrete & Continuous Random Variables
▪ A discrete random variable is a random variable with a finite or
countably infinite range. Its values are obtained by counting
▪ A continuous random variable is a random variable with an interval
(either finite or infinite) of real numbers for its range. Its values are
obtained by measuring
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Examples of Discrete & Continuous Random Variables
▪ Discrete random variables:
✓Number of scratches on a surface
✓Proportion of defective parts among 100 tested
✓Number of transmitted bits received in error
✓Number of common stock shares traded per day
▪ Continuous random variables:
✓Electrical current and voltage
✓Physical measurements, e.g., length, weight, time, temperature,
pressure
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Discrete vs. Continuous Random Variables
▪ Number when rolling a dice





discrete
discrete
discrete
▪ The sum when rolling two dice
▪ Number of children in a family
counting
x
0
2
4
6
▪ Time of running 10 km
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10
continuous
continuous
continuous
▪ Amount of sugar in a juice
▪ Height of females
 measure




x
0
2
4
6
8
10
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Example 3.1 | Flash Recharge Time
› The time to recharge the flash is tested in three
cellphone cameras (Table 3: sample space and
associated probabilities))
o The probability that a camera passes the
test is 0.8 and the cameras perform
independently.
o Random Variable X denotes number of
cameras that pass the test
o Cameras are independent, hence:
𝑃(𝑝𝑝𝑝) = (0.8)(0.8)(0.8) = 0.521
› The last column shows the values of 𝑋
assigned to each outcome of the experiment.
Table 3.1 Camera Flash Tests
Camera #
1
2
3
Probability X
Pass
Pass
Pass
0.512
3
Fail
Pass
Pass
0.128
2
Pass
Fail
Pass
0.128
2
Fail
Fail
Pass
0.032
1
Pass
Pass
Fail
0.128
2
Fail
Pass
Fail
0.032
1
Pass
Fail
Fail
0.032
1
Fail
Fail
Fail
0.008
0
Sum
1.000
Sec 3.1 Probability Distributions and Mass Functions
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