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)263 ‫اإلحصاء و االحتماالت (عرض‬
Probability and Statistics
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Course code MSC 263
Class Days Sunday, Wednesday
Credit hours 3hrs
Instructor A. Sultan
Email [email protected]
Site www.freewebs.com/amsultan_52
Dr. Ahmed M. Sultan
)263 ‫اإلحصاء و االحتماالت (عرض‬
Probability and Statistics
• Course Grading Policy
– Quiz1 wk4 5 pts
– MT wk9 15 pts
– Quiz2 wk11 5 pts
– Hw 15 pts
– Project 10 pts
– Final 50 pts
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Dr. Ahmed M. Sultan
Topics
• Review of probability theory
– Random Variables
– Conditional probability and conditional expectation
• The analysis of variance
– Introduction
– Single factor ANOVA
• Simple linear regression and correlation
– Introduction
– The simple linear regression model
• Estimation model parameters
• Inferences about the slope parameters
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Dr. Ahmed M. Sultan
Topics (Cont.)
• Multivariate regression analysis
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When to use multivariate regression
Control variables
Interpreting coefficients
Goodness of fit (R squared statistic)
• The exponential distribution and the Poisson
process
• Queueing theory
– The M/M/1 queue
• Steady state probabilities
• Some performance measures
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Dr. Ahmed M. Sultan
Topics (Cont.)
– The M/M/m queue
• Steady state probabilities
• Some performance measures
– The M/M/1/K queue
• Steady state probabilities
• Some performance measures
• Discrete event simulation
– Generating pseudo random numbers
• Congruential methods for generating pseudo random numbers
• Composite generators
• Statistical tests for goodness of fit
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Dr. Ahmed M. Sultan
Topics (Cont.)
• Generating stochastic variables
– The inverse transformation method
– Sampling from continuous probability distribution
• Data manipulation in MINITAB
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Recording and transforming variables
Graphs and charts
Scatter plots
Histograms
Box plots and other charts
Cross tabulation
Dr. Ahmed M. Sultan
References
• Devore, J. “Probability and statistics for
engineering and sciences”
• Andrews Willing “A short introduction to
queueing theory”
• Banks, et. al. “Discrete event simulation”
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Dr. Ahmed M. Sultan
Review of probability theory
1. Laws of probability
DEFINITION
If an event E occurs m times in an n trial
experiment, then the probability P(E) is
defined as:
i.e the experimrnt is repeated infinitely
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Dr. Ahmed M. Sultan
• e.g
In case of flipping a coin, the longer the
experiment is repeated the closer will be the
estimate to P(H) (or P(T)) to the theortical
value of 0.5
0≤P(E) ≤1
P(E)=0 … E is impossible
P(E)=1 … E is certain (sure)
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Dr. Ahmed M. Sultan
HW
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In a study to correlate senior year high school students scores
in mathematics and enrollment in engineering colleges a 1000
students were surveyed:
400 have studied mathematics
Engineering enrollment shows that of the 1000 seniors:
150 have studied mathematics
29 have not
Determine the probability of:
a. A student who studied mathematics is enrolled in engineering
b. A student who neither studies mathematics nor enrolled in
engineering
c. A student is not studying engineering
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Dr. Ahmed M. Sultan
1.1 Addition law of probability
EUF … Union of E and F
EF … Intersection of E and F
If EF = ɸ, E and F are mutually exclusives or
disjointed (occurrence of one precludes the
other)
• Addition law
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Dr. Ahmed M. Sultan
Example
Rolling a die
S={1,2,3,4,5,6} sample space
P(1)= P(2)= P(3)= P(4)= P(5)= P(6)=1/6
Define
E={1,2,3, or 4}
F={3, 4, or 5}
EF={3,4}
P(E)= P(1)+ P(2)+ P(3)+ P(4)=4/6=2/3
P(F)=3/6=1/2
P(EF)=2/6=1/3
P(EUF)=P(E)+P(F)-P(EF)
=2/3+1/2-1/3=5/6
Which is intuitively clear since
EUF={1,2,3,4,5}
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Dr. Ahmed M. Sultan
HW
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A fair die is tossed twice. E and F represent the
outcomes of the two tosses. Compute the
following probabilities
Sum of E and F is 11
Sum of E and F is even
Sum of E and F is odd and greater than 3
E is even less than 6 and F is odd greater than 1
E is graeter than 2 and F is less than 4
E is 4 and sum of E and F is odd
Dr. Ahmed M. Sultan
1.2 Conditional Probability
• The conditional probability of an event E is the
probability that the event will occur given the
knowledge that an event F has already occurred.
This probability is written P(E|F), notation for the
probability of E given F.
• In the case where events E and F are independent
(where event F has no effect on the probability of
event E), the conditional probability of event E
given event F is simply the probability of event E,
that is P(E).
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Dr. Ahmed M. Sultan
…(Cont.)
• If events E and F are not independent, then the probability
of the intersection of E and F (the probability that both
events occur) is defined by
P(E F) = P(F)P(E|F).
• From this definition, the conditional probability P(E|F) is
easily obtained by dividing by P(F):
• P(E|F) = P(EF) / P(F) , P(F) > 0
• Note: This expression is only valid when P(F) is greater than
0.
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Dr. Ahmed M. Sultan
Example
In rolling a die, what is the probability that the
outcome is 6, given that the rolling turned up
an even number
Solution
E={6}, F={2,4,6} thus
P(E|F)=P(EF)/P(F)=P(E)/P(F)=(1/6)/(1/2)=1/3
Note that P(EF)=P(E) because E is a subset of F
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Dr. Ahmed M. Sultan
Example
Ninety percent of flights depart on time. Eighty percent of
flights arrive on time. Seventy-five percent of flights depart
on time and arrive on time.
(a) You are meeting a flight that departed on time. What is the
probability that it will arrive on time?
(b) You have met a flight, and it arrived on time. What is the
probability that it departed on time?
(c) Are the events, departing on time and arriving on time,
independent?
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Dr. Ahmed M. Sultan
Solution
Denote the events,
A = { arriving on time} ,
D = {departing on time} .
P{A} = 0.8, P{D} = 0.9, P{AD} = 0.75.
(a) P{A I D} = P{AD} / P{D} = 0.75 / 0.9 = 0.8333
(b) P{D I A}= P{AD} / P{A} = 0.75 / 0.8 = 0.9375
(c) Events are not independent because
P{AI D} ≠ P{A}, P{DI A} ≠ P{D}, P{AD} ≠ P{A}P{D}.
Actually, anyone of these inequalities is sufficient to prove that A and D are
dependent. Further, we see that P{AI D} > P{A} and P{D I A} > P {D}. In other words,
departing on time increases the probability of arriving on time, and vise versa. This
perfectly agrees with our intuition.
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Dr. Ahmed M. Sultan
HW
In the example of rolling a die if given that the
outcome is less than 6, determine
probability of getting :
a. an even number
b. an odd number larger than 1.
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Dr. Ahmed M. Sultan
HW
• You can toss a fair coin up to 7 times. You will
win 1000 SR if three tails appear before a
head is encountered. What are your chances
of wining?
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Dr. Ahmed M. Sultan