10 Transformations
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10 Transformations

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10 Transformations 10 Transformations Presentation Transcript

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