Statistics 110

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Statistics 110

• Balaji S. Srinivasan
• Week 4 Numerical Examples,
• Conditional Probability, Independence

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Syllabus
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Last Week

• Expectation of g(X)
• Properties
• Mean Variance

• Multivariate RVs
• Data Frames
• Multinomial
• Multivariate Normal

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Today

• Review
• Expectation
• Random Vectors

• Numerical Examples
• Analytical Calculation
• Compare to R Simulation

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Mean and Variance
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Random Vectors
A random vector is a function which maps simple
events to vectors
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Random Vectors
We tabulate random vectors in data frames
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Discrete Random Vectors
Data frames implicitly specify joint marginal
pmfs
Joint PMF (Empirical Frequencies)
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Discrete Random Vectors
Data frames implicitly specify joint marginal
pmfs
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Multinomial
Key Discrete Random Vector
What is the probability of getting 10 red, 5
blue, 7 green if we draw 22 balls?
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Random Vectors
What if our random vector has continuous elements?
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Continuous RVs
Again we can build up a data frame...
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Continuous RVs
Data frames implicitly specify joint marginal
pdfs
samples
Each pair of points is a sample from a joint PDF
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Continuous RVs
Practice
Theory
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Data Frames
Data frames as random vectors...
abstract but powerful!
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Numerical Examples
Ok. Lets solidify with some examples
First review of R and RVs
Then a day in the life at Stanford
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Preliminaries
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And Geometric...
week4_lecture10.R
dbinom plot of pmf
rbinom hist of random deviates
pbinom plot of cdf
qbinomplot of quantiles
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Mean Variance
Geometry of Mean and Variance
Pop. Variance and SD
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Important
Theory and Sample differ!
To understand this we need sampling
distributions...
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Example 1
Waiting Times
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Example 1
Waiting Times
He sees that 10 people were served in 5 minutes
and decides to (crudely) model w/ an exponential
distro...
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Waiting Times
Now he can answer some questions
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Some comments...
Why not the binomial?
Assume 1000 people served over course of night.
What is the probability that at least 1 person
waits 3 mins?
In this case we could have used binomial (exact
result) but usually for large N/small p we have
overflow problems w/ exact computation (leading
to Poisson)
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Example 2
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Example 2
Riding on Campus
Albert wishes to ride his bike without getting a
ticket. He observes that the police only ticket a
subset of all arrivals.
Arrivals are normally distributed around 11am w/
SD of 5 mins. Police start and stop ticketing so
that they catch 90 of dangerous felons...
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Example 2
Riding on Campus
Arrivals are normally distributed around 11am w/
SD of 5 mins. Police start and stop ticketing so
that they catch 90 of dangerous felons...
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Example 2
Riding on Campus
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Example 2
Riding on Campus
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Today

• Review
• Expectation
• Random Vectors

• Numerical Examples
• Analytical Calculation
• Compare to R Simulation