1. Quiz
• Pick up quiz on your way in.
• Start at 1pm
• Finish at 1:10pm
• Bonus: no homework this week.
Homework help sessions -> review
sessions.
Tuesday, 16 February 2010
4. Transformations
Distribution Change of
function variable
technique technique
Much more useful for 2+ rvs
Tuesday, 16 February 2010
5. Remember: it’s the sample
space that changes, not
the probability
Tuesday, 16 February 2010
6. Distribution function
technique
Y = u(X)
1. Find region in X such that g(X) < y
2. Find cdf with integration
3. Find pdf by differentiating
Tuesday, 16 February 2010
7. Finish the proof
X = Uniform(-1, 1)
Y = X2
P(Y < y) = P(X2< y) = ...
Tuesday, 16 February 2010
8. Change of variables
If Y = u(X), and
v is the inverse of u, X = v(Y)
then
fY(y) = fX(v(y)) |v’(y)|
Transformation must
have an inverse!
Tuesday, 16 February 2010
9. Your turn
X = Uniform(-1, 1)
Y= X2
Can you use the change of variables
technique here? Why/why not? If not,
how could you modify X to make it
possible?
Tuesday, 16 February 2010
10. Your turn
X ~ Exponential(β)
Y = exp(X)
Find fY(y). Does y have a named
distribution?
Tuesday, 16 February 2010
11. Relationship to uniform
Important connection between the
uniform and every other random variable
through the cdf.
Tuesday, 16 February 2010
12. Uniform to any rv
IF
Y ~ Uniform(0, 1)
F a cdf
THEN
X= F -1(Y) is a rv with cdf F(x)
(Assume F strictly increasing for simplicity)
Tuesday, 16 February 2010
13. Any rv to uniform
IF
X has cdf F
Y = F(X)
THEN
Y ~ Uniform(0, 1)
(Assume F strictly increasing for simplicity)
Tuesday, 16 February 2010
15. F (x) = P (X ≤ x)
discrete F (x) = f (t)
tx
x
continuous F (x) = f (t)dt
−∞
Tuesday, 16 February 2010
16. f (x)
Integrate Differentiate
F(x)
Tuesday, 16 February 2010
17. lim F (x) = 0
x→−∞
lim F (x) = 1
x→∞
monotone increasing
right-continuous
Tuesday, 16 February 2010
18. lim F (x) = 0
x→−∞
lim F (x) = 1
x→∞
monotone increasing
right-continuous
Tuesday, 16 February 2010
19. Using the cdf
P (a X ≤ b) = F (b) − F (a)
Exact computation
Tables
Computer
Tuesday, 16 February 2010
20. Exact computation
For some distributions we can write the
cdf in closed form. For example: the
exponential distribution has cdf:
1−e −λx
, x ≥ 0,
F (x; λ) =
0, x 0.
Tuesday, 16 February 2010
21. Exact computation
Many, however cannot:
http://en.wikipedia.org/wiki/
Binomial_distribution
http://en.wikipedia.org/wiki/
Gamma_distribution
integrate 1/(sqrt(2 pi)) e ^ (-t^2 / 2) from -
infinity to x
Tuesday, 16 February 2010
22. Closed form
Neither the CDF of the normal distribution
nor erf can be expressed in terms of finite
additions, subtractions, multiplications,
and root extractions, and so both must be
either computed numerically or otherwise
approximated.
i.e. can’t be expressed in closed form
http://mathworld.wolfram.com/NormalDistribution.html
Tuesday, 16 February 2010
23. Two approaches
Look up the number in a table.
Problem: you need a table for every
combination of the parametes
Use a computer.
Problem: you need a computer
Tuesday, 16 February 2010
25. Standard normal
Fortunately, for the normal distribution, we
can convert any random variable with a
normal distribution to a standard normal.
This means we only need one table for
any possible normal distribution. (For
other distributions there will be multiple
tables, and typically you will have to pick
one with similar values to your example).
Tuesday, 16 February 2010
26. 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
Tuesday, 16 February 2010
27. Using the tables
Column + row = z
Find: Φ(2.94), Φ(-1), Φ(0.01), Φ(4)
Can also use in reverse: For what value
of z is P(Z z) = 0.90 ? i.e. What is Φ-1(0.90)?
Find: Φ-1(0.1), Φ-1(0.5), Φ-1(0.65), Φ-1(1)
Tuesday, 16 February 2010
28. Next time
Using the computer instead
Generating random values and simulation
Tuesday, 16 February 2010
