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### 18 Normal Cont

1. 1. Stat310 Normal distribution Hadley Wickham Thursday, 19 March 2009
2. 2. 1. Another summer opportunity 2. Recap 3. Standard normal 4. Sums of normals 5. Chi-square distribution Thursday, 19 March 2009
3. 3. VIGRE research project Over summer \$5,000 Work with me (or anyone else in stats department) Email me if you’re interested Thursday, 19 March 2009
4. 4. Recap What is the pdf of the normal distribution? What is the mgf? What is the mean and variance? How do you create a standard normal? What is the pdf and mgf of the gamma distribution? Thursday, 19 March 2009
5. 5. Standard normal If X ~ Normal(μ, σ2), and Z = (X - μ) / σ Then: Z ~ Normal(0, 1) = standard normal How can we show this? (What if X isn’t normal?) Thursday, 19 March 2009
6. 6. Using the standard normal These days, you don’t need to use the standard normal, you can just use a computer. But it’s useful for exams, and more importantly, it’s how statisticians tend to think about the normal distribution Use plot to give rough estimate. Thursday, 19 March 2009
7. 7. Using the tables Column + row = z Find: Φ(2.94), Φ(-1), Φ(0.01), Φ(4) Can also use in reverse: For what value of z is P(Z < z) = 0.90 ? i.e. What is Φ-1(0.90)? Find: Φ-1(0.1), Φ-1(0.5), Φ-1(0.65), Φ-1(1) Thursday, 19 March 2009
8. 8. 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 Thursday, 19 March 2009
9. 9. Example The time it takes me to bike to school is normally distributed with mean 10 and standard deviation 4. What is the probability it takes me more than 20 minutes to bike to school? What time should I leave so that I have 95% chance of getting to class by 1pm? Thursday, 19 March 2009
10. 10. Example What’s the probability I take a negative amount of time to get to school? Is the distribution of my bike times really normal? Thursday, 19 March 2009
11. 11. Conﬁdence interval I’d like to create a 95% conﬁdence interval for my biking time. i.e. I want to ﬁnd a and b such that P(a < X < b) = 0.95. How many ways are there to construct this interval? Generally want to ﬁnd the interval with the shortest length. How can I do that? Thursday, 19 March 2009
12. 12. Sums of normals Normal(μi, σi Let Xi ~ 2), independent Y = c1X1 + c2X2 + … + cnXn What is the distribution of Y? (What is the mean and variance of Y?) How can we work it out? Thursday, 19 March 2009
13. 13. Example If Z1, Z2, Z3 are independent standard normals, what is the distribution of: Z1 - Z3 Z1 + Z2 + Z3 (Z1 + Z2 + Z3)/3 Thursday, 19 March 2009
14. 14. CLT prequel If X1, X2, …, Xn are iid N(μ, σ2) What is the distribution of their average? Thursday, 19 March 2009
15. 15. Another special distribution If X ~ Normal(μ, σ2), and V = (X - μ)2 / σ2 = Z2 Then V ~ χ2(1) Thursday, 19 March 2009
16. 16. Chi-squared Skipped over it in Chapter 3 Special case of the gamma distribution, when θ = 2 and α = r / 2 (r an integer) Mean = r, Variance = 2r r is called degrees of freedom Thursday, 19 March 2009
17. 17. http://en.wikipedia.org/wiki/Chi-square_distribution Thursday, 19 March 2009
18. 18. Sums Let Z1, Z2, …, Zn be iid N(0, 1) W = Z12 + Z22 + … + Zn2 Then W ~ χ2(n) This is going to be useful when we try to estimate the variance Thursday, 19 March 2009
19. 19. Why? Thursday, 19 March 2009
20. 20. 20 15 ● ● ● ● ● ● time 10 ● ● ● ● 5 0 1 2 3 4 5 day Thursday, 19 March 2009
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24. 24. 70 60 50 40 count 30 20 10 0 0 5 10 15 20 time Thursday, 19 March 2009
25. 25. 20 ● ● ● ● 15 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● time 10 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 5 ● ● ● 0 1.0 1.5 2.0 2.5 3.0 3.5 4.0 week Thursday, 19 March 2009