Transcript
Hadley Wickham
Stat310Transformations
Wednesday, 10 February 2010
Explorations in Statistics Research
http://www.stat.berkeley.edu/~summer/
7 day workshop in Boulder, Colorado
Travel + room & board covered
Large datasets, real research problems, and data visualisation.
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1. Test info
2. Normal distribution (theory)
3. Transformations
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Feb 18. 80 minute in class test. 4 questions.
One double sided sheet of notes.
Covers everything up to Feb 16: probability and random variables/distributions. See website for exactly what you should know.
Approximately half applied (working with real problems) and half theoretical (working with mathematical symbols).
Test
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ExpectationsPoints will be awarded for fully converting a word problem into a mathematical problem.
You should be able to differentiate & integrate polynomials and exponentials and apply the chain rule.
I will supply random mathematical facts and tables of probabilities (if needed).
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Note sheet
Much of the usefulness of a note sheet is the process of making it.
You want to condense everything we have covered. Pull out ongoing themes. Make tables. Use colour!
Not useful: a photocopy of someone else’s notes, a verbatim copy of the textbook
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The normal distribution
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f(x)
0.0
0.1
0.2
0.3
0.4
−10 −5 0 5 10
f(x)
0.0
0.1
0.2
0.3
0.4
−10 −5 0 5 10
f(x)
0.0
0.1
0.2
0.3
0.4
−10 −5 0 5 10
f(x)
0.0
0.1
0.2
0.3
0.4
−10 −5 0 5 10
N(-2, 1) N(5, 1)
N(0, 1)
N(0, 16)N(0, 4)
f(x)
0.0
0.1
0.2
0.3
0.4
−10 −5 0 5 10
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f(x) =1√2π
e−(x−µ)2
2σ2
Is this a valid pdf?
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Wolfram alpha
integrate 1/(sigma sqrt(2 pi)) e ^ (-(x- mu)^2 / (2(sigma^2))) from -infinity to infinity
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Not good enough :(Let’s do it by hand...
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M(t) = eµt+ 12 σ2t2
A few tricks + lots of algebra
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Your turn
If X ~ Normal(μ, σ2), use the mgf to confirm that the mean and variance are what you expect.
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Cheating...
d/dt e^(mu*t + 1/2 sigma^2 t^2) at t = 0
d^2/dt^2 e^(mu*t + 1/2 sigma^2 t^2) at t = 0
d^2/dz^2 exp(mu*z + 1/2 sigma^2 z^2) at z = 0
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Transformations
If X ~ Normal(μ, σ2), and Y = a(X + b)
Y ~ Normal(b + μ, a2σ2)
If a = -μ and b = 1/σ, we often write
Z = (X - μ) / σZ ~ Normal(0, 1) = standard normal
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Example
Let X ~ Normal(5, 10)
What is P(3 < X < 8) ?
Learn how to answer that question on Thursday.
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P (Z < z) = Φ(z)
P (−1 < Z < 1) = 0.68P (−2 < Z < 2) = 0.95P (−3 < Z < 3) = 0.998
Φ(−z) = 1− Φ(z)
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Transformations
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Discrete
Let X be a discrete random variable with pmf f as defined above.
Write out the pmfs for:
A = X + 2 B = 3*X C = X2
x -5 0 5 10 20
f(x) 0.2 0.1 0.3 0.1 0.3
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Continuous
Let X ~ Unif(0, 1)
What are the distributions of the following variables?
A = 10 X
B = 5X + 3
C = X2
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0.0
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0.6
0.8
1.0
0.2 0.4 0.6 0.8
X ~ Uniform(0, 1)
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0.0
0.2
0.4
0.6
0.8
1.0
0.2 0.4 0.6 0.8
X ~ Uniform(0, 1)
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0.00
0.02
0.04
0.06
0.08
0.10
2 4 6 8
10X
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0.00
0.05
0.10
0.15
0.20
4 5 6 7
5X + 3
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0.0
0.2
0.4
0.6
0.8
1.0
0.2 0.4 0.6 0.8
X ~ Uniform(0, 1)
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0
5
10
15
20
0.2 0.4 0.6 0.8
X2
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0.0
0.5
1.0
1.5
0.2 0.4 0.6 0.8
sqrt(X)
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Next time
Computing probabilities
Simulation
No reading, BUT GOOD OPPORTUNITY TO REVIEW CURRENT MATERIAL
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