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5.2 Joint Continuous Distributions
Anna Karlin Most slides by Alex Tsun
5.2 Joint Continuous Distributions Anna Karlin Most slides by Alex - - PowerPoint PPT Presentation
5.2 Joint Continuous Distributions Anna Karlin Most slides by Alex - - PowerPoint PPT Presentation
5.2 Joint Continuous Distributions Anna Karlin Most slides by Alex Tsun recap Joint PDFs (Example 1) R R Joint PDFs (Example 1) R R Joint PDFs (Example 1) R R Joint PDFs (Example 1) R R Joint PDFs (Example 1) R R Joint PDFs
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Joint PDFs (Example 1)
R R
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Joint PDFs (Example 1)
R R
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Joint PDFs (Example 1)
R R
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Joint PDFs (Example 1)
R R
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Joint PDFs (Example 1)
R R
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Joint PDFs (Example 1)
R R
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Random Picture
probability students Definition of Expectation
The “Normal” Distribution The “Gaussian” Distribution
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Joint PDFs (Example 2)
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Joint PDFs (Example 2)
1 y=x
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Joint PDFs (Example 2)
1 y=x
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Joint PDFs (Example 2)
1 y=x
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Joint PDFs (Example 2)
1 y=x
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Joint PDFs (Example 2)
1 y=x
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Joint PDFs (Example 2)
1 y=x
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Joint PDFs (Example 2)
1 y=x
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Joint PDFs (Example 2)
1 y=x
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Joint PDFs (Example 2)
1 y=x
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Joint PDFs (Example 2)
1 y=x
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Joint PDFs (Example 2)
1 y=x
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Joint PDFs (Example 2)
1 y=x
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END PIC
Alex Tsun Joshua Fan
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5.3 Law of Total Expectation
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Agenda
- Conditional Expectation
- Law of Total Expectation (LTE)
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Conditional Expectation
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E(X|A) = X
x∈Range(X)
xPr(X = x|A)
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Law of Total Expectation
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Law of Total Expectation : Application
System that fails in step i independently with probability p X # steps to fail E(X) ?
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Let A be the event that system fails in first step.
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Law of Total Expectation : Application
System that fails in step i independently with probability p X # steps to fail E(X) ?
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Let A be the event that system fails in first step. E(X) = E(X|A)Pr(A) + E(X|A)Pr(A) = p + (1 + E(X))(1 − p) = 1 + (1 − p)E(X) E(X) = 1 p
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Linearity of expectation applies
To conditional expectation too!! E(X+ Y | A) = E(X | A) + E(Y | A) E(aX + b | A)= a E(X | A) + b
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Law of total Expectation (RV version)
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Problem
The number of people who enter an elevator on the ground floor is a Poisson random variable with mean 10. If there are N floors above the ground floor, and if each person is equally likely to get off at any one of the N floors, independently of where the others get off, compute the expected number of stops that the elevator will make before discharging all the passengers.
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