class02-1-handouts

# class02-1-handouts - PSTAT 120B Probability& Statistics...

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Unformatted text preview: PSTAT 120B - Probability & Statistics Class # 02-1- Sampling distributions Jarad Niemi University of California, Santa Barbara 5 April 2010 Jarad Niemi (UCSB) Sampling distributions 5 April 2010 1 / 23 Class overview Announcements Announcements Homework 1 due today, either on the front table when class ends or 3-4pm in South Hall 5521 (Rachev room) R code available in a GauchoSpace folder Jarad Niemi (UCSB) Sampling distributions 5 April 2010 2 / 23 Class overview Day’s Goals Goals Goal What is a statistic? Some important statistics What is the distribution for Y = 1 n ∑ n i =1 Y i when Y i iid ∼ N ( μ,σ 2 )? What is the distribution for ¯ Y- μ S / √ n where S is the sample standard deviation? New distributions t-distribution χ 2-distribution F-distribution Jarad Niemi (UCSB) Sampling distributions 5 April 2010 3 / 23 Sampling distributions Statistics Statistics Definition A statistic is a function of the observable random variables in a sample and known constants. Examples Sample mean Y = 1 n ∑ n i =1 Y i Sample variance S 2 = 1 n- 1 ∑ n i =1 ( Y- Y ) 2 Sample maximum Y ( n ) = max( Y 1 , Y 2 ,..., Y n ) Sample minimum Y (1) = min( Y 1 , Y 2 ,..., Y n ) Inter-quartile range R = Y ( n )- Y (1) Jarad Niemi (UCSB) Sampling distributions 5 April 2010 4 / 23 Sampling distributions Sampling distributions Sampling distributions Definition The sampling distribution is the probability distribution of a statistic. Jarad Niemi (UCSB) Sampling distributions 5 April 2010 5 / 23 Normal sample mean Distribution derivation Theorem Suppose Y 1 , Y 2 ,..., Y n iid ∼ N ( μ,σ 2 ) , Y = 1 n ∑ n i =1 Y i has a normal distribution with mean μ and variance σ 2 / n. Proof. Use the method of moment-generating functions: E e t Y = E e t 1 n ∑ n i =1 Y i = E e ∑ n i =1 t n Y i = E Q n i =1 e t n Y i (properties of exponentials) = Q n i =1 E e t n Y i (independent) = Q n i =1 exp μ t n + ( t n ) 2 σ 2 2 = exp μ t n + ( t n ) 2 σ 2 2 n = exp μ t + t 2 σ 2 / n 2 So Y ∼ N ( μ,σ 2 / n ). Jarad Niemi (UCSB) Sampling distributions 5 April 2010 6 / 23 Normal sample mean Simulation example From the previous slide: Y ∼ N ( μ,σ 2 / n ) and P μ- σ √ n < Y < μ- σ √ n = 68% ....
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class02-1-handouts - PSTAT 120B Probability& Statistics...

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