class02-1-handouts

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

Info iconThis preview shows pages 1–8. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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 Days 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 ( , 2 )? What is the distribution for Y- S / n where S is the sample standard deviation? New distributions t-distribution 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 ( , 2 ) , Y = 1 n n i =1 Y i has a normal distribution with mean and variance 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 t n + ( t n ) 2 2 2 = exp t n + ( t n ) 2 2 2 n = exp t + t 2 2 / n 2 So Y N ( , 2 / n ). Jarad Niemi (UCSB) Sampling distributions 5 April 2010 6 / 23 Normal sample mean Simulation example From the previous slide: Y N ( , 2 / n ) and P - n < Y < - n = 68% ....
View Full Document

Page1 / 23

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

This preview shows document pages 1 - 8. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online