- #1
squenshl
- 479
- 4
Give an example to show that if not assuming independence of X1, X2, ..., Xn it is possible to show that Var(1/n * sum from k = 1 to n of Xk) >> [tex]\sigma^2/n[/tex]
Assuming independence means that the probability of one event occurring does not affect the probability of another event occurring. In other words, the two events are considered to be unrelated and do not influence each other.
One example of assuming independence is flipping a coin. The outcome of one coin flip does not affect the outcome of another coin flip. Each flip is considered to be independent of the others.
If independence is not assumed, statistical analysis becomes more complex and may require different methods. This is because the assumption of independence is often necessary for certain statistical tests to be valid.
An example of this is conducting a survey on the relationship between smoking and lung cancer, but not considering other factors such as genetics or exposure to secondhand smoke. If these factors are not taken into account, the results may suggest a strong correlation between smoking and lung cancer when in reality, the relationship may not be as strong if independence is assumed.
There are various methods for accounting for lack of independence in statistical analysis, such as using multilevel or longitudinal models, or conducting sensitivity analyses to assess the impact of the lack of independence on the results.