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09 Normal Trans

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09 Normal Trans

1. 1. Stat310 Transformations Hadley Wickham Wednesday, 10 February 2010
2. 2. Explorations in Statistics Research http://www.stat.berkeley.edu/~summer/ 7 day workshop in Boulder, Colorado Travel + room & board covered Large datasets, real research problems, and data visualisation. Wednesday, 10 February 2010
3. 3. 1. Test info 2. Normal distribution (theory) 3. Transformations Wednesday, 10 February 2010
4. 4. Test Feb 18. 80 minute in class test. 4 questions. One double sided sheet of notes. Covers everything up to Feb 16: probability and random variables/distributions. See website for exactly what you should know. Approximately half applied (working with real problems) and half theoretical (working with mathematical symbols). Wednesday, 10 February 2010
5. 5. Expectations Points will be awarded for fully converting a word problem into a mathematical problem. You should be able to differentiate & integrate polynomials and exponentials and apply the chain rule. I will supply random mathematical facts and tables of probabilities (if needed). Wednesday, 10 February 2010
6. 6. Note sheet Much of the usefulness of a note sheet is the process of making it. You want to condense everything we have covered. Pull out ongoing themes. Make tables. Use colour! Not useful: a photocopy of someone else’s notes, a verbatim copy of the textbook Wednesday, 10 February 2010
7. 7. The normal distribution Wednesday, 10 February 2010
8. 8. 0.4 0.4 N(-2, 1) N(5, 1) 0.3 0.3 0.2 0.2 f(x) f(x) 0.1 0.1 0.0 0.0 −10 −5 0 5 10 0.4 −10 −5 0 5 10 0.3 N(0, 1) 0.2 f(x) 0.1 0.0 0.4 −10 −5 0 5 10 0.4 N(0, 4) N(0, 16) 0.3 0.3 0.2 0.2 f(x) f(x) 0.1 0.1 0.0 0.0 −10 −5 0 5 10 −10 −5 0 5 10 Wednesday, 10 February 2010
9. 9. 1 (x−µ) − 2σ2 2 f (x) = √ e 2π Is this a valid pdf? Wednesday, 10 February 2010
10. 10. Wolfram alpha integrate 1/(sigma sqrt(2 pi)) e ^ (-(x- mu) ^2 / (2(sigma^2))) from -inﬁnity to inﬁnity Wednesday, 10 February 2010
11. 11. Not good enough :( Let’s do it by hand... Wednesday, 10 February 2010
12. 12. 1 2 2 M (t) = e µt+ 2 σ t A few tricks + lots of algebra Wednesday, 10 February 2010
13. 13. Your turn σ If X ~ Normal(μ,2),use the mgf to conﬁrm that the mean and variance are what you expect. Wednesday, 10 February 2010
14. 14. Cheating... d/dt e^(mu*t + 1/2 sigma^2 t^2) at t = 0 d^2/dt^2 e^(mu*t + 1/2 sigma^2 t^2) at t =0 d^2/dz^2 exp(mu*z + 1/2 sigma^2 z^2) at z=0 Wednesday, 10 February 2010
15. 15. Transformations If X ~ Normal(μ, σ2), and Y = a(X + b) Y ~ Normal(b + μ, a 2σ2) If a = -μ and b = 1/σ, we often write Z = (X - μ) / σ Z ~ Normal(0, 1) = standard normal Wednesday, 10 February 2010
16. 16. Example Let X ~ Normal(5, 10) What is P(3 < X < 8) ? Learn how to answer that question on Thursday. Wednesday, 10 February 2010
17. 17. P (Z < z) = Φ(z) Φ(−z) = 1 − Φ(z) P (−1 < Z < 1) = 0.68 P (−2 < Z < 2) = 0.95 P (−3 < Z < 3) = 0.998 Wednesday, 10 February 2010
18. 18. Transformations Wednesday, 10 February 2010
19. 19. Discrete 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 Wednesday, 10 February 2010
20. 20. Continuous Let X ~ Unif(0, 1) What are the distributions of the following variables? A = 10 X B = 5X + 3 C= X2 Wednesday, 10 February 2010
21. 21. 1.0 X ~ Uniform(0, 1) 0.8 0.6 0.4 0.2 0.0 0.2 0.4 0.6 0.8 Wednesday, 10 February 2010
22. 22. 1.0 X ~ Uniform(0, 1) 0.8 0.6 0.4 0.2 0.0 0.2 0.4 0.6 0.8 Wednesday, 10 February 2010
23. 23. 0.10 10X 0.08 0.06 0.04 0.02 0.00 2 4 6 8 Wednesday, 10 February 2010
24. 24. 0.20 5X + 3 0.15 0.10 0.05 0.00 4 5 6 7 Wednesday, 10 February 2010
25. 25. 1.0 X ~ Uniform(0, 1) 0.8 0.6 0.4 0.2 0.0 0.2 0.4 0.6 0.8 Wednesday, 10 February 2010
26. 26. X2 20 15 10 5 0 0.2 0.4 0.6 0.8 Wednesday, 10 February 2010
27. 27. sqrt(X) 1.5 1.0 0.5 0.0 0.2 0.4 0.6 0.8 Wednesday, 10 February 2010
28. 28. Next time Computing probabilities Simulation No reading, BUT GOOD OPPORTUNITY TO REVIEW CURRENT MATERIAL Wednesday, 10 February 2010