10 Transformations

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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

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