Upcoming SlideShare
×

# 10 Transformations

327 views
285 views

Published on

0 Likes
Statistics
Notes
• Full Name
Comment goes here.

Are you sure you want to Yes No
• Be the first to comment

• Be the first to like this

Views
Total views
327
On SlideShare
0
From Embeds
0
Number of Embeds
4
Actions
Shares
0
4
0
Likes
0
Embeds 0
No embeds

No notes for slide

### 10 Transformations

1. 1. Stat310 Transformations Hadley Wickham Monday, 16 February 2009
2. 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. 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. 4. Exponential • Derivation • Moment generating function Monday, 16 February 2009
5. 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. 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
7. 7. Transformations Monday, 16 February 2009
8. 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. 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. 10. Transformations Distribution Change of function variable technique technique Monday, 16 February 2009
11. 11. Distribution function technique X = Unif(0, 1) Y = X2 P(Y < y) = P(X2< y) = P(X < √y) ... Monday, 16 February 2009
12. 12. Your turn X ~ Exponential(θ) Y = log(X) Find fY(y). Does y have a named distribution? Monday, 16 February 2009
13. 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. 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. 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. 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