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16 Sequences

1. 1. Stat310 Sequences of rvs Hadley Wickham Wednesday, 17 March 2010
2. 2. Major’s day 2:30-4:30pm Today Oshman Engineering Design Kitchen Come along and talk to me (or Rudy Guerra) if you’re interested in becoming a stat major Wednesday, 17 March 2010
3. 3. Assessment Test model answers online tonight (hopefully) Usual help session tonight 4-5pm. Wednesday, 17 March 2010
4. 4. 1. Sequences 2. Limits 3. Chebyshev’s theorem 4. The law of large numbers 5. The central limit theorem Wednesday, 17 March 2010
5. 5. Sequences 1 variable: X 2 variables: X, Y ... n variables: X1, X2, X3, ..., Xn Wednesday, 17 March 2010
6. 6. Sequences Xi ~ Normal(μi, σi) Xi ~ Normal(μ, σi) Xi ~ Normal(μi, σ) Xi ~ Normal(μ, σ) Almost always assume that the Xi’s are independent. In the last case they are also identically distributed. Wednesday, 17 March 2010
7. 7. iid = independent & identically distributed Wednesday, 17 March 2010
8. 8. Your turn Xi are iid N(0, 2). What is E(X30)? What is Var(X2001)? What is Cor(X10, X11)? Cor(X1, X1000)? Wednesday, 17 March 2010
9. 9. n n E( Xi ) = E(Xi ) i i n n V ar( ai Xi ) = 2 ai V ar(Xi ) i i If what is true? n n E( Xi ) = E(Xi ) i i If what is true? Wednesday, 17 March 2010
10. 10. Limits Typically will deﬁne some function of n ¯ random variables, e.g. Xn ¯ What happens to Xn when n → ∞? Why? Because often it will converge, and we can use this to approximate results for any large n. Wednesday, 17 March 2010
11. 11. New notation If xn → 0, and n is big, we can say xn ≈ 0. If Xn → Z, Z ~ N(0, 1), and n is big, we can say Xn ~ . N(0,1). Read as approximately distributed. Other ways to write it Wednesday, 17 March 2010
12. 12. N go o od lim art Chebyshev it ing st -b p ut oin a t 1 P (|X − µ| Kσ) ≥ 1 − 2 K 1 P (|X − µ| Kσ) ≤ 2 K For K 0 Wednesday, 17 March 2010
13. 13. Your turn How can you put this in words? 1 P (|X − µ| Kσ) ≤ 2 K Wednesday, 17 March 2010
14. 14. The probability of being more than K standard deviations 80 away from the mean is less than one over K squared. 60 (For K 0) 1 K2 40 20 0 2 4 6 8 10 K Wednesday, 17 March 2010
15. 15. (For K 1) 1.0 0.8 0.6 1 K2 0.4 0.2 0.0 2 4 6 8 10 K Wednesday, 17 March 2010
16. 16. Your turn How does this compare to the normal distribution? Compare the probability of being less than 1, 2 and 3 standard deviations away from the mean given by Chebychev and what we know about the normal. Wednesday, 17 March 2010
17. 17. 1.0 0.8 0.6 variable value cheby norm 0.4 0.2 0.0 2 4 6 8 10 x Wednesday, 17 March 2010
18. 18. LLN Law of large numbers X1, X2, ..., Xn iid. n ¯ Xn = Xi i There are ﬁve ways to write the result. Wednesday, 17 March 2010
19. 19. What does it mean? As we collect more and more data, the sample mean gets closer and closer to the true mean. Not that surprising! But note that we didn’t make any assumptions about the distributions Wednesday, 17 March 2010
20. 20. CLT Central limit theorem. The distribution of a mean is normal when gets big. Wednesday, 17 March 2010
21. 21. Approximation This implies that if n is big then ... Wednesday, 17 March 2010
22. 22. Reading Section 4.1 Focus on the general ideas and the deﬁntions Wednesday, 17 March 2010