09 Normal Trans

Stat310 Transformations

Hadley Wickham
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.


1. Test info
2. Normal distribution (theory)
3. Transformations


Test
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).


Expectations
Points 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).


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


The normal
distribution


0.4 0.4
N(-2, 1) N(5, 1)
0.3 0.3

0.2 0.2
f(x)

f(x)
0.1 0.1

0.0 0.0
−10 −5 0 5 10
0.4 −10 −5 0 5 10

0.3 N(0, 1)

0.2
f(x)

0.1

0.0
0.4 −10 −5 0 5 10
0.4

N(0, 4) N(0, 16)
0.3 0.3

0.2 0.2
f(x)

f(x)
0.1 0.1

0.0 0.0
−10 −5 0 5 10 −10 −5 0 5 10


1 (x−µ)
− 2σ2
2

f (x) = √ e
2π

Is this a valid pdf?

Wolfram alpha

integrate 1/(sigma sqrt(2 pi)) e ^ (-(x- mu)
^2 / (2(sigma^2))) from -inﬁnity to inﬁnity


Not good enough :(
Let’s do it by hand...


1 2 2
M (t) = e µt+ 2 σ t

A few tricks + lots of algebra

Your turn

σ
If X ~ Normal(μ,2),use the mgf to
conﬁrm that the mean and variance are
what you expect.


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


Transformations
If X ~ Normal(μ, σ2), and Y = a(X + b)
Y ~ Normal(b + μ, a 2σ2)

If a = -μ and b = 1/σ, we often write
Z = (X - μ) / σ
Z ~ Normal(0, 1) = standard normal


Example

Let X ~ Normal(5, 10)
What is P(3 < X < 8) ?
Learn how to answer that question on
Thursday.


P (Z < z) = Φ(z)
Φ(−z) = 1 − Φ(z)

P (−1 < Z < 1) = 0.68
P (−2 < Z < 2) = 0.95
P (−3 < Z < 3) = 0.998

Transformations


Discrete
x -5 0 5 10 20
f(x) 0.2 0.1 0.3 0.1 0.3

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


Continuous
Let X ~ Unif(0, 1)
What are the distributions
of the following variables?
A = 10 X
B = 5X + 3
C= X2


1.0
X ~ Uniform(0, 1)

0.8

0.6

0.4

0.2

0.0

0.2 0.4 0.6 0.8


0.10
10X

0.08

0.06

0.04

0.02

0.00

2 4 6 8


0.20
5X + 3

0.15

0.10

0.05

0.00

4 5 6 7


X2

20

15

10

5

0

0.2 0.4 0.6 0.8


sqrt(X)

1.5

1.0

0.5

0.0

0.2 0.4 0.6 0.8


Next time

Computing probabilities
Simulation
No reading, BUT GOOD OPPORTUNITY
TO REVIEW CURRENT MATERIAL


09 Normal Trans

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