This is page 2 of 2, my question is at the end of this page
Hello,
I am verifing an exact value of the 2nd moment E(R^2) of a sample range (R) from the standard normal distribution. To let you clearly know what is the question that I am confronted with. Please see the attachments for details.
To save your time, you can only read the highlighted part on page 1 to get the context. My question is at the end of page 2.
Your help is highly appreciated !
Thanks a lot in advance.
DHB10 from China, PRC
Last edited by DHB10; 02-03-2012 at 01:22 AM.
This is page 2 of 2, my question is at the end of this page
Last edited by DHB10; 02-03-2012 at 12:31 AM.
Hello,
My another related question:
Because we are dealing with the random variable XY, for sure, both X and Y follow the normal distribution respectively, Is the random variable Z= XY follows the bivariate normal distribution ? If yes, we can get the joint probability density funcction f(x,y) of the random variable XY as shown in equation (3) upstairs.
Thanks a lot in advance.
Last edited by DHB10; 02-03-2012 at 01:30 AM.
Are you talking about X= X(1) and Y=X(n)? Because in that case X and Y don't have normal distributions.
DHB10 (02-03-2012)
are order statistics and in general they do not follow the same distribution as the original sample. So you cannot say they have a marginal/jointly normal distribution.
To obtain the joint pdf is not hard. For example, in your case you can follow the following arguments:
To obtain , it is equivalent to:
1. Having 0 sample smaller than
2. Having 1 sample at
3. Having n - 2 sample between
4. Having 1 sample at
5. Having 0 sample larger than
And hence therefore using the multinomial arguments
DHB10 (02-03-2012)
Hello, BGM & Dason,
Many thanks for your help.
To BGM, in your equation upstairs, are f(x) and F(x) the pdf and cdf of the standard normal distribution respectively?
Thanks
DHB10 from China, PRC
The equation BGM posted gives the joint distribution for the min/max of a sample of size n for any distribution where F is the cdf and f is the pdf. In your case you would want to use the pdf and cdf of the standard normal.
Hello, Dason,
Many thanks for your explanation!
I am going to use both your way to compute the E(R^2).
DHB10 from CN
@DHB10, I would suggest you look at this article:
http://www.sciencedirect.com/science...67715207002143
See Equation (12).
DHB10 (02-03-2012)
Hello, Dragan,
Nice to hear you again and thanks for your information, I am going to see the article you mentioned upstairs.
Thanks again.
DHB10 from China.
Hello, BGM,
Many thanks for your help.
Meanwhile, I would like to show something interested in the following attachments. Although when n = 2, the computation result of E(R^2) is correct in my previous, at least, is the same as the author's version, however, it seems that it is coincidently correct, because according to some evidences, the probablity density function of the random variable Z = XY in this context is not what I used when computing E(R^2) in my previous post, it should have been the one shown in the following attachment ( page 3 of 3) based on the evidences shown in the following attachements.
DHB10 from China, PRC
This is page 2 of 3
Last edited by DHB10; 02-03-2012 at 03:50 PM.
DHB10 (02-03-2012)
This is page 2 of 2
Last edited by DHB10; 02-03-2012 at 09:35 AM.
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