Give an example to show that if not assuming independence of

Thread starter squenshl
Start date May 8, 2011
Tags

Example Independence

AI Thread Summary

Assuming non-independence among random variables X1, X2, ..., Xn can lead to significant variance in their average. When all variables are equal, such as X1 = X2 = ... = Xn, the variance of their average becomes equal to the variance of a single variable, which is σ². This results in Var((1/n) * sum from k = 1 to n of Xk) equating to σ², demonstrating that the variance does not diminish with increasing n. Consequently, the variance of the average can exceed σ²/n, contradicting the assumption of independence. This example illustrates the critical impact of variable dependence on variance calculations.

May 8, 2011

squenshl

Give an example to show that if not assuming independence of X₁, X₂, ..., X_n it is possible to show that Var(1/n * sum from k = 1 to n of X_k) >> \sigma^2/n

Physics news on Phys.org

May 9, 2011

g_edgar

What do you get if X_1 = X_2 = ..., which is sort of the EXTREME example of non-independence?

May 9, 2011

squenshl

Since X₁ = X₂ = ... = X_n.
This implies that Var((1/n)*nX₁) = n²\sigma^2/n² = \sigma^2 >> \sigma^2/n

Last edited: May 9, 2011

Thread 'Deductive proof in logic formal systems'

I was reading documentation about the soundness and completeness of logic formal systems. Consider the following $$\vdash_S \phi$$ where ##S## is the proof-system making part the formal system and ##\phi## is a wff (well formed formula) of the formal language. Note the blank on left of the turnstile symbol ##\vdash_S##, as far as I can tell it actually represents the empty set. So what does it mean ? I guess it actually means ##\phi## is a theorem of the formal system, i.e. there is a...

View full post »

Thread 'Onto set mapping is the surjective set mapping, and into injective?'

The textbook is being fine. I asked the forum for some introduction to topology, and decided to start with Simmon`s. This naive question is due to ignorance of the words into and onto, which I don't distinguish in Spanish. A quick browsing sugests I'm right.

View full post »