Joint Used to Show lack of Correlation?

[FactChecker]

If the $x_1,x_2,...x_n$ are the possible values of the population, the mean of the population is not defined to be $\frac{ \sum_{i=1}^n x_i}{n}$.

Unless they are uniformely distributed, each with a probability 1/n.

 

[WWGD]

If the $x_1,x_2,...x_n$ are the possible values of the population, the mean of the population is not defined to be $\frac{ \sum_{i=1}^n x_i}{n}$.

Precisely. But this is the definition used in the book I browsed. I understand that the x_i are scaled by f(x_i). I may have missed a section where the author states the assumption that these variables are uniformly distributed.Edit: I remember that a random sample from a population is a collection $\{X_1,...,X_n \}$ of independent, I.D random variables.

 

[FactChecker]

But this is the definition used in the book I browsed.

Just to be careful with words, I wouldn't call that the definition. I would call the $\sum f(x_i)*x_i$ the definition. In the case of a uniform distribution, this turns out to be $\sum_{i=1}^n 1/n *x_i$

 

[Stephen Tashi]

Precisely. But this is the definition used in the book I browsed.

You need to read the book carefully enough to understand the difference between a population mean and a sample mean. (The conceptual structure of mathematical statistics is extremely sophisticated. For example, what is the definition of a "statistic"?)

The fact that the sample mean of observations $x_1,x_2,...x_n$ is defined to be $\frac{ \sum_{i=1}^n x_i}{n}$ has no assumption or implication about the distribution from which the samples are drawn. If the sample values are $\{1,1,2,2,2,2,3\}$ there is nothing in the definition of sample mean that says you treat the values $\{1,2,3\}$ as if they are uniformly distributed.

 

[WWGD]

Please see attached , page 5 of Doug C. Montgomery's Applied stats. Couldn't make picture clearer but hopefully clear-enough.

 

[Stephen Tashi]

It's too blurred. What are you trying to convey by quoting it?

 

[WWGD]

Please see attached , page 5 of Doug C. Montgomery's Applied stats. Couldn't make picture clearer but hopefully clear-enough.

Seriously, I am not making it up, I am using the definition from the reference I cited. I am on my phone now, will look it up on web and if it is there, I will attach it.

 

[WWGD]

It's too blurred. What are you trying convey by quoting it?

I am trying to show that the definition of population mean used in the book is precisely the same as the arithmetic mean, for finite population s.

 

[Stephen Tashi]

I am trying to show that the definition of population mean used in the book is precisely the same as the arithmetic mean, for finite population s.

There is nothing in those words that assumes the values in the population are uniformly distributed. The expression $\frac{\sum_{i=1}^n x_i }{n}$ makes no assumption that each $x_i$ is a different value.

There is also nothing in those words that assumes the sampling procedure must be to give each member of the population the same probability of being included in the sample.

Where assumptions about the distribution enter the picture is when we want to prove theorems about the behavior of the sample mean as a random variable. A typical theorem would assume a sample is taken from a population in a particular manner (e.g. "random sampling without replacement"). However the definition of sample mean is not a theorem about the sample mean.