Register to reply 
Statistics, calculate the distribution 
Share this thread: 
#1
Mar712, 11:32 AM

P: 297

1. The problem statement, all variables and given/known data
Assume [itex] z_1, ..., z_m[/itex] are iid,[itex] z_i = μ+\epsilon_i [/itex] [itex] \epsilon_i] [/itex]is N(0,σ^2) Show that f(z; μ) = g([itex] \bar{z}[/itex]; μ)h(z) where h(·) is a function not depending on μ. 2. Relevant equations 3. The attempt at a solution Now z is normal distributed with mean my and variance sigma^2 [itex] f(z,\mu) = \frac{1}{\sigma^2 \sqrt{2 \pi}} e^{\frac{(z\mu)^2}{2 \sigma^2}} [/itex] f(z; μ) = [itex]\prod_{i=1}^m \frac{1}{\sigma^2 \sqrt{2 \pi}} e^{\frac{(z_i\mu)^2}{2 \sigma^2}} [/itex] [itex] f(\bf{z},\mu) = (\frac{1}{\sigma^2 \sqrt{2 \pi}})^m \prod_{i=1}^m e^{\frac{(z_i\mu)^2}{2 \sigma^2}} [/itex] but how do I go from here to f(z; μ) = g([itex] \bar{z}[/itex]; μ)h(z) And am I in the right track? 


#2
Mar712, 12:53 PM

Sci Advisor
HW Helper
Thanks
P: 4,945

RGV 


#3
Mar712, 04:14 PM

P: 297

Thanks
I'm not sure what the notation mean, but I assumed g was a function of the sample mean and mu. But if it is the pdf of the sample mean. The pdf of [itex]\bar{z} [/itex] is the pdf to the normal distribution with mean [itex]\mu] [/itex] and variance [itex] \sigma^2/m [/itex] So [itex] g(\bar{z}) = \frac{1}{(\sigma^2/m )\sqrt{2 \pi}} e^{\frac{(\bar{z}\mu)^2}{2 (\sigma^2/m)}} [/itex] But I'm not sure where to go from here. 


#4
Mar712, 05:04 PM

Sci Advisor
HW Helper
Thanks
P: 4,945

Statistics, calculate the distribution
[tex] g = \frac{1}{\sqrt{2\pi}\sigma/\sqrt{n}}\exp{\left[\left(\frac{z_1+z_2+\cdots+z_n}{n}\mu \right)^2/(2 \sigma^2 /n)\right]}. [/tex] You are supposed to show that f(z,μ)/g does not have μ in it. PS: your expressions for the normal distributions are a bit wrong: you should have[itex] \frac{1}{\sigma \sqrt{2 \pi}}, \text{ not } \frac{1}{\sigma^2 \sqrt{2 \pi}}[/itex] in front. RGV 


#5
Mar812, 01:44 PM

P: 297

Thanks for all the help.
To those who want to see the rest of the calculation it is in the attachment. 


Register to reply 
Related Discussions  
Simulation, distribution, statistics.  Set Theory, Logic, Probability, Statistics  2  
Statistics Distribution  Calculus & Beyond Homework  2  
Statistics  normal distribution  Calculus & Beyond Homework  4  
Statistics: Fdistribution  Calculus & Beyond Homework  1  
Statistics  Normal distribution  Introductory Physics Homework  2 