Download Probability and Statistics

Probability & Statistics
• Outline Probability and Statistics
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2 types of probability
Rules of probability
Statistical Independence
Expected Value
Normal Distributions
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2 Types of Probability
• Subjective
– Probability estimate based on what a person believes or
experiences
– “I think there is a 60% chance of rain tomorrow.”
• Objective
– Probabilities that can be stated before or a priori the
occurrence of an event
– Roll of a fair dice
– Flip of a fair coin
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Fundamentals and Rules of Probability
• Rules of probability
1. 0 < P(A) < 1
2. SPi = 1
3. P(A or B) = P(A) + P(B), for mutually exclusive A & B
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Mutual Exclusivity
• Only one event can occur at a time
A
B
• Addition rule
• P(A) + P(B) = P(A OR B)
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Probability Types
• Marginal
– Probability of a single event
– e.g., P(A) = 0.1
• Joint
– Probability of more than one event
– e.g., P(A and B) = 0.2
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Example of a Joint Probability
• Probability of two non-mutually exclusive events
occurring
A
B
• Shaded area is a joint probability
• General addition rule
• P(A or B) = P(A) + P(B) - P(A and B)
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Independence
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Successive events that do not affect one another
e.g., flipping a coin
P(H and {then} T) = P(H)*P(T)
In the case of dependent events
P(A and {then} B) = P(B)*P(A|B)
General rule of multiplication
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Conditional Probability
• Probability that an event will occur given that
another event has already occurred
• e.g., weather forecasts, given thunder what is the
probability of rain
• P(A|B)
• Reads probability of A given B
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Independent vs. Dependent Events
• Independent Events
– P(A|B) = P(A)
– P(A and B) = P(A)*P(B)
• Dependent Events
– P(A|B) = P(A and B) / P(B)
– P(A and B) = P(B)*P(A|B)
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Bayes’ Theorem
• Famous statistician/mathematician
• Created relationship for dependent events
P(A|B) = P(A and B) / P(B)
• Total probability law
P(B) = P(B|A)*P(A) + P(B|not A)*P(not A)
• Used to update probabilistic forecasts, e.g.,
weather forecasts
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Example of Bayes’ Theorem
• Given, A and B are dependent events
P(A and B) = 0.2, P(B) = 0.4
Calculate P(A|B):
P(A|B) = P(A and B) / P(B) = 0.2 / 0.4 = 0.5
• Given, A and B are independent events
Calculate P(A|B):
P(A|B) = P(A)*P(B) / P(B) = P(A)
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Expected Value
• Mean of the probability distribution of a random
variable (RV)
• E(x) = Sx*P(x) e.g.,
x
0
1
2
3
P(x)
0.1
0.2
0.3
0.4
Sx*P(x) = E(x) =
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x*P(x)
0.0
0.2
0.6
1.2
2.0
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Normal Distribution
• Common probability distribution
• e.g., height, weight, age, sum of two dice rolled 1,000
times, etc.
Normal Distribution
0.4
0.35
0.3
P(x)
0.25
0.2
0.15
0.1
0.05
0
0
1
2
3
4
5
6
7
8
x
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Normal Distribution
Normal Distribution
0.4
mean = 4, std. dev. = 1
0.35
0.3
P(x)
0.25
0.2
-1 std. dev. +1 std. dev.
0.15
0.1
68% of values
0.05
0
0
1
2
3
4
5
6
7
8
x
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Mean and Standard Deviation
• Most common statistics used
• Mean or expected value
E(x) = SxiP(xi)
xi

m= n
• Standard deviation
s(x) = [S [xi - E(x)]2P(xi)]0.5
s(x) = [S [xi - m]2/n-1]0.5
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Z-Scores
• Standard Z-score
• Measures the number of standard deviations away from
the mean
• Calculated as such:
x  mean
Z
SD
• Look up Z value in table to find probability
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