How is the sample mean defined?
What is the expectation of the sample mean equal to?
What must be true of the variables that are added together to compute the sample mean for the expression you have written down to hold?
What is the variance of the sample mean equal to?
What must be true of of the variables that were added together to compute the sample mean for the expression that you have written for the sample variance to be valid?
The sample mean is a statistic that is calculated as follows:
$$ \overline{X} = \frac{1}{N} \sum_{i=1}^N X_i $$
where each of the $X_i$ in the sum is an identically distributed random variable. The sample mean is an unbiased estimator for the expectation of the random variable. As the sample mean is computed by adding together random variables it is, however, a random variable.
The expectation of the sample mean is:
<aside> 💡 $\mathbb{E}(\overline{X}) = \mathbb{E}(X)$
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where $\mathbb{E}(X)$ is the expectation of the random variables from which the sample mean was computed. The proof of this result is explained in the following video:
https://www.youtube.com/watch?v=qfThUCzX4g0
Notice that the result for the expectation above holds as long as the random variables that are added together are all identical. Importantly, the random variables that are added together do not need to be independent.
If all the $X_i$ from which the sample mean are computed are independent as well as identical then the variance of the sample mean is given by:
<aside> 💡 $\textrm{var}(\overline{X}) = \frac{\textrm{var}(X)}{N}$
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The proof of this result is explained in the following video:
https://www.youtube.com/watch?v=GDP4VeNfUhg&t=403s
When a sample mean is computed from more random variables it becomes a more precise estimator for the expectation of the distribution.
<aside> 📌 SUMMARY: Any sample mean computed by adding together random variables is itself a random variable. We can write expressions for the expectation and variance of the sample mean in terms of the expectation and variance of the random variables from which the sample mean was computed.
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