1. Stat310 Transformations
Hadley Wickham
Monday, 16 February 2009
2. 1. Recap
2. Exponential derivation
3. Transforming random variables
1. Distribution function technique
2. Change of variables technique
4. Interesting properties of cdf
Monday, 16 February 2009
3. Recap
f (x) = cx 0 < x < 10
• What is the cdf?
• What must c be for f to be a pdf?
• What is P(2 < X < 8)?
Monday, 16 February 2009
4. Exponential
• Derivation
• Moment generating function
Monday, 16 February 2009
5. Your turn
Let Y be the amount of time until I make a
mistake on the board. Assume Y ~
Exp(10) (i.e. I make 10 mistakes per hour).
If I go for 30 minutes without making a
mistake, what’s the probability I go for 40
minutes without making a mistake?
i.e. What is P(Y > 40 | Y > 30) ? How
does it compare to P(Y > 10)?
Monday, 16 February 2009
6. Memorylessness
• In general, if Y is exponential
• P(Y > y + a | Y > y ) = P(Y > a)
• Can you prove that?
• No memory
Monday, 16 February 2009
8. Example
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
Monday, 16 February 2009
9. Continuous
Let X ~ Unif(0, 1)
What are the distributions
of the following variables?
A = 10 X
B = 5X + 3
C= X2
Monday, 16 February 2009
10. Transformations
Distribution Change of
function variable
technique technique
Monday, 16 February 2009
11. Distribution function
technique
X = Unif(0, 1)
Y = X2
P(Y < y) = P(X2< y) = P(X < √y)
...
Monday, 16 February 2009
12. Your turn
X ~ Exponential(θ)
Y = log(X)
Find fY(y). Does y have a named
distribution?
Monday, 16 February 2009
13. 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)|
Monday, 16 February 2009
14. Your turn
X ~ Exponential(θ). Y = log(X).
What is fY(y)?
X ~ Uniform(0, 10). Y = X2.
What is fY(y)?
Monday, 16 February 2009
15. Theorem 3.5-1
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)
Monday, 16 February 2009
16. Theorem 3.5-2
IF
X has cdf F
Y = F(X)
THEN
Y ~ Uniform(0, 1)
(Assume F strictly increasing for simplicity)
Monday, 16 February 2009
