Let Y1, Y2, …be a sequence of random variables with E(Yi)=μand V(Yi)=σi^2 . Notice that the σi^2

step 1 For this exercise, we have E of Y bar sub n is equivalent to 1 over n, sigma i equals 1 to n, gi

Let Y1, Y2, …be a sequence of random variables with...

Let $Y_{1}, Y_{2}, \ldots$ be a sequence of random variables with $E\left(Y_{i}\right)=\mu$ and $V\left(Y_{i}\right)=\sigma_{i}^{2} .$ Notice that the $\sigma_{i}^{2}$ 's are not all equal. a. What is $E\left(\bar{Y}_{n}\right) ?$ b. What is $V\left(\bar{Y}_{n}\right) ?$ c. Under what condition (on the $\sigma_{i}^{2}$ 's) can Theorem 9.1 be applied to show that $\bar{Y}_{n}$ is a consistent estimator for $\mu ?$

Let $Y_{1}, Y_{2}, \ldots$ be a sequence of random variables with $E\left(Y_{i}\right)=\mu$ and $V\left(Y_{i}\right)=\sigma_{i}^{2} .$ Notice that the $\sigma_{i}^{2}$ 's are not all equal.
a. What is $E\left(\bar{Y}_{n}\right) ?$
b. What is $V\left(\bar{Y}_{n}\right) ?$
c. Under what condition (on the $\sigma_{i}^{2}$ 's) can Theorem 9.1 be applied to show that $\bar{Y}_{n}$ is a consistent estimator for $\mu ?$

Key Concepts

Consistency of an Estimator

An estimator is consistent if it converges in probability to the true parameter as the sample size increases. For the sample mean of independent but non-identically distributed random variables to be consistent, the variance of the sample mean must converge to zero as the number of observations increases. This typically requires a condition on the individual variances—such as those being uniformly bounded—to ensure that their cumulative effect diminishes with larger sample sizes.

Linearity of Expectation

This principle states that the expected value of a sum of random variables is equal to the sum of their individual expected values, irrespective of whether the random variables are independent. In the context of a sample mean, this concept guarantees that if each random variable has the same mean, then the expected value of the sample mean is equal to that common mean.

Variance of Independent Random Variables

For independent random variables, the variance of the sum is the sum of the variances. When considering the sample mean, the variance of this estimator is obtained by summing the variances of the individual random variables and then dividing by the square of the number of terms. This method highlights the impact of non-identical variances on the overall variability of the estimator.

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