To determine the approximate distribution of the random variable

$\frac{1}{n} \sum_{i=1}^n (X_i - \mu),$

we can analyze it using the Central Limit Theorem (CLT). Here's the breakdown:

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Step 1: Analyze the random variable

The given random variable is

$\frac{1}{n} \sum_{i=1}^n (X_i - \mu).$

Since $\mu = \mathbb{E}[X_i]$, the term $X_i - \mu$ represents the deviation of $X_i$ from its mean. The expectation and variance of this random variable are:

1. Mean: By linearity of expectation: [ \mathbb{E}\left[\frac{1}{n} \sum_{i=1}^n (X_i - \mu)\right] = \frac{1}{n} \sum_{i=1}^n \mathbb{E}[X_i - \mu] = \frac{1}{n} \cdot 0 = 0. ]

2. Variance: Since $X_1, X_2, \ldots, X_n$ are independent and have variance 1: $\text{Var}\left(\frac{1}{n} \sum_{i=1}^n (X_i - \mu)\right) = \frac{1}{n^2} \sum_{i=1}^n \text{Var}(X_i - \mu) = \frac{1}{n^2} \cdot n \cdot 1 = \frac{1}{n}.$

Thus, the random variable has mean 0 and variance $\frac{1}{n}$.

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Step 2: Central Limit Theorem Application

As $n \to \infty$, the Central Limit Theorem states that the sum (or average) of a large number of independent and identically distributed random variables converges in distribution to a normal (Gaussian) distribution.

• The scaled sum: $\sqrt{n} \left(\frac{1}{n} \sum_{i=1}^n (X_i - \mu)\right)$ converges in distribution to a standard normal distribution $\mathcal{N}(0, 1)$ as $n \to \infty$.

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Step 3: Convergence of the Original Random Variable

The random variable $\frac{1}{n} \sum_{i=1}^n (X_i - \mu)$ therefore converges to a normal distribution with mean 0 and variance $\frac{1}{n}$.

Hence, as $n$ becomes "very large," the random variable $\frac{1}{n} \sum_{i=1}^n (X_i - \mu)$ is approximately:

$\mathcal{N}\left(0, \frac{1}{n}\right).$

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Final Answer:

The distribution of $\frac{1}{n} \sum_{i=1}^n (X_i - \mu)$ is approximately:

$\mathcal{N}\left(0, \frac{1}{n}\right).$

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5 Related Questions:

1. How does the variance of the sample mean change as $n$ increases?
2. Why is the Central Limit Theorem applicable in this case?
3. What happens if the $X_i$ random variables are not independent?
4. How does this result connect to the Law of Large Numbers?
5. What are practical examples where this approximation is useful?

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Tip:

When working with sums or averages of random variables, always check conditions like independence, identical distributions, and finite variance to determine if the CLT applies.