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# 06 Moments

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### 06 Moments

1. 1. Stat310 Moments Hadley Wickham Saturday, 30 January 2010
2. 2. Engineer Your Career Monday, February 15 7:00 PM - 8:30 PM McMurtry Auditorium Find out what you can do with a degree in engineering from a panel of successful Rice engineering graduates who have gone into a variety of professions. (Plus get dessert!) http://engineering.rice.edu/EventsList.aspx? EventRecord=13137 Saturday, 30 January 2010
3. 3. Homework Due today. From now on, if late, put in Xin Zhao’s mail box in the DH mailroom. Another one due next Thursday Buy a stapler Use ofﬁcial name Saturday, 30 January 2010
4. 4. 1. Finish off proof 2. More about expectation 3. Variance and other moments 4. The moment generating function 5. The Poisson distribution 6. Feedback Saturday, 30 January 2010
5. 5. Proof, continued Saturday, 30 January 2010
6. 6. Expectation of a function Saturday, 30 January 2010
7. 7. Expectation Expectation is a linear operator: Expectation of a sum = sum of expectations (additive) Expectation of a constant * a function = constant * expectation of function (homogenous) Expectation of a constant is a constant. T 2.6.2 p. 95 Saturday, 30 January 2010
8. 8. Your turn Write (or recall) the mathematical description of these properties. Work in pairs for two minutes. (Extra credit this week is to prove these properties) Saturday, 30 January 2010
9. 9. Moments The ith moment of a random variable is deﬁned as E(X i) = μ' . The ith central i moment is deﬁned as E[(X - E(X)) i] = μ i The mean is the ________ moment. The variance is the ________ moment. Saturday, 30 January 2010
10. 10. Name Symbol Formula 1 mean μ μ'1 2 variance σ2 μ2 = μ'2 - μ 2 3 skewness α3 μ3 /σ3 4 kurtosis α4 μ4 /σ4 Saturday, 30 January 2010
11. 11. 3 4 var =1 0.6 0.5 0.4 skew = 0 0.3 kurt = 3.4 0.2 0.1 0.0 5 6 0.6 0.5 0.4 0.3 0.2 0.1 0.0 2 4 6 8 2 4 6 8 Saturday, 30 January 2010
12. 12. 0.4 1.2 1.6 mean = 4 0.6 0.5 0.4 skew = 0 0.3 kurt ≈ 2.5 0.2 0.1 0.0 2.6 2.8 3.6 0.6 0.5 0.4 0.3 0.2 0.1 0.0 2 4 6 8 2 4 6 8 2 4 6 8 Saturday, 30 January 2010
13. 13. −1.83 −1.03 −1.02 −0.91 0.8 0.6 0.4 0.2 0.0 −0.61 −0.21 0.21 0.91 0.8 0.6 0.4 0.2 0.0 1.02 1.83 0.8 0.6 mean ≈ 4 0.4 var = 1.3 0.2 0.0 2 4 6 8 2 4 6 8 2 4 6 8 2 4 6 8 Saturday, 30 January 2010
14. 14. 1.00 1.46 1.59 mean = 4 0.5 0.4 skew = 0 0.3 var ≈ 4 0.2 0.1 0.0 1.87 2.05 2.26 0.5 0.4 0.3 0.2 0.1 0.0 2 4 6 8 2 4 6 8 2 4 6 8 Saturday, 30 January 2010
15. 15. mgf The moment generating function (mgf) is Mx(t) = E(eXt) (Provided it is ﬁnite in a neighbourhood of 0) Why is it called the mgf? (What happens if you differentiate it multiple times). Useful property: If MX(t) = MY(t) then X and Y have the same pmf. Saturday, 30 January 2010
16. 16. Plus, once we’ve got it, it can make it much easier to ﬁnd the mean and variance Saturday, 30 January 2010
17. 17. Expectation of binomial (take 2) Figure out mgf. (Random mathematical fact: binomial theorem) Differentiate & set to zero. Then work out variance. Saturday, 30 January 2010
18. 18. Your turn Compute mean and variance of the binomial. Remember the variance is the 2nd central moment, not the 2nd moment. Saturday, 30 January 2010
19. 19. Poisson 3.2.2 p. 119 Saturday, 30 January 2010
20. 20. Poisson distribution X = Number of times some event happens (1) If number of events occurring in non- overlapping times is independent, and (2) probability of exactly one event occurring in short interval of length h is ∝ λh, and (3) probability of two or more events in a sufﬁciently short internal is basically 0 Saturday, 30 January 2010
21. 21. Poisson X ~ Poisson(λ) Sample space: positive integers λ ∈ [0, ∞) Saturday, 30 January 2010
22. 22. 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 Saturday, 30 January 2010
23. 23. Example On average, a small amount of radioactive material emits ten alpha particles every ten seconds. If we assume it is a Poisson process, then: What is the probability that no particles are emitted in 10 seconds? Make sure to set up mathematically. Saturday, 30 January 2010
24. 24. mgf, mean & variance Random mathematical fact. Compute mgf. Compute mean & 2 nd moment. Compute variance. Saturday, 30 January 2010
25. 25. Next week Repeat for continuous variables. Make absolutely sure you have read 2.5 and 2.6. (hint hint) Saturday, 30 January 2010
26. 26. Feedback Saturday, 30 January 2010