Hadley Wickham Stat310 Transformations Wednesday, 10 February 2010
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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.
Wednesday, 10 February 2010
1. Test info
2. Normal distribution (theory)
3. Transformations
Wednesday, 10 February 2010
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
Wednesday, 10 February 2010
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
Wednesday, 10 February 2010
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
Wednesday, 10 February 2010
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
Wednesday, 10 February 2010
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
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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0.0
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
Wednesday, 10 February 2010
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