Final Answer:
Yes, Xˉ is a consistent estimator for E[X].
Step-by-step explanation:
The Strong Law of Large Numbers states that Xˉ→E[X] as n→∞ when Xn is i.i.d. This means that the sample mean will converge to the population mean as the sample grows infinitely large. In other words, as the sample size increases, the sample mean becomes a more accurate estimate of the population mean. Therefore, Xˉ is a consistent estimator for E[X] because it converges to the true population mean as the sample size increases.
In mathematical terms, for any ε > 0, we have lim┬(n→∞) P(|Xˉ - E[X]| > ε) = 0. This means that the probability of the sample mean differing from the population mean by more than ε approaches zero as the sample size grows infinitely large. Hence, Xˉ is a consistent estimator for E[X] as it satisfies the definition of consistency by converging in probability to the true parameter value.
In conclusion, Xˉ is a consistent estimator for E[X] because it satisfies the conditions of consistency by converging to the true population mean as the sample size increases, as guaranteed by the Strong Law of Large Numbers.