08 Continuous

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08 Continuous

  1. 1. Stat310 Continuous variables Hadley Wickham Tuesday, 3 February 2009
  2. 2. 1. Notes about the exam 2. Finish off Poisson 3. Introduction to continuous variables 4. The uniform distribution Tuesday, 3 February 2009
  3. 3. Exam • Exam structure • Grading tomorrow • Purpose of notes • Question 1 - most of you managed to get it (eventually) - at least 3 different ways • Question 2 & 4 - did really well • Question 3 - more of a struggle Tuesday, 3 February 2009
  4. 4. Poisson distribution X = Number of times some event happens If number of events occurring in non- overlapping times is independent, and Probability of exactly one event occurring in short interval of length h is ∝ λh, and Probability of two or more events in a sufficiently short internal is basically 0 Then X ~ Poisson(λ) Tuesday, 3 February 2009
  5. 5. Examples Number of calls to a switchboard Number of eruptions of a volcano Number of alpha particles emitted from a radioactive source Number of defects in a roll of paper Tuesday, 3 February 2009
  6. 6. λ=1 λ=2 0.35 0.25 0.30 0.20 0.25 0.15 0.20 f(x) f(x) 0.15 0.10 0.10 0.05 0.05 0.00 0.00 0 5 10 15 20 0 5 10 15 20 x x λ=5 λ = 20 0.12 0.15 0.10 0.08 0.10 f(x) f(x) 0.06 0.04 0.05 0.02 0.00 0.00 0 5 10 15 20 0 5 10 15 20 x x Tuesday, 3 February 2009
  7. 7. What is λ? • What is the sample space of X? • Let’s start by looking at the mean and variance of X. • How? Tuesday, 3 February 2009
  8. 8. What is λ? • λ is the mean rate of events per unit time. • If you change the unit of time from 1 to t, you’ll expect λt events - another Poisson process/distribution • ie. if X ~ Poisson(λ), and Y = tX, then Y ~ Poisson(λt) Tuesday, 3 February 2009
  9. 9. Example • A small amount of radioactive material emits one alpha particle on average every second. If we assume it is a Poisson process, then: • How many particles would be emitted ever minute, on average? • What is the probability that no particles are emitted in 10 seconds? Tuesday, 3 February 2009
  10. 10. Continuous random variables Tuesday, 3 February 2009
  11. 11. Continuous r.v. • Sample space is the real line • Mathematical tools: more differentiation + integration • Same vocabulary, slightly different definitions • New distributions Tuesday, 3 February 2009
  12. 12. Intuition Imagine you have a spinner which is equally likely to point in any direction. Let X be the angle the spinner points. What is P(X ∈ [0, 90]) ? What is P(X ∈ [270, 90]) ? What is P(X ∈ [70, 98]) ? What is the general formula? What is P(X = 90) ? Tuesday, 3 February 2009
  13. 13. Cumulative distribution function x F (x) = P (X ≤ x) = f (t)dt −∞ b P (X ∈ [a, b]) = f (x)dx = F (b) − F (a) a P (X = a) = P (x = [a, a]) = F (a) − F (a) = 0 Tuesday, 3 February 2009
  14. 14. f (x) For continuous x, f(x) is a probability density function. Not a probability! Tuesday, 3 February 2009
  15. 15. f (x) Integrate Differentiate F(x) Tuesday, 3 February 2009
  16. 16. Conditions f (x) ≥ 0 ∀x ∈ R f (x) = 1 R Tuesday, 3 February 2009
  17. 17. Questions? Is f(x) < 1 for all x? What do those conditions imply about F(x)? Tuesday, 3 February 2009
  18. 18. E(u(X)) = u(x)f (x)dx R MX (t) = e f (x)dx tx R Tuesday, 3 February 2009
  19. 19. The discrete uniform Assigns probability uniformly in an interval [a, b] of the real line X ~ Uniform(a, b) What are F(x) and f(x) ? b f (x)dx = 1 a Tuesday, 3 February 2009
  20. 20. Intuition X ~ Unif(1, b) What do you expect the mean of X to be? What about the variance? Tuesday, 3 February 2009
  21. 21. a+b E(X) = 2 (b − a) 2 V ar(X) = 12 Tuesday, 3 February 2009
  22. 22. Question X ~ Unif(0, 1) Y = 10 X What is the distribution of Y? How does the variance of Y compare to the variance of X? Tuesday, 3 February 2009

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