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Can you tell me if my understanding about the CLT is correct? Maybe it's just matter of notation.

The classical CLT states:

Let $X_1,...,X_i,...,X_n$ be a sequence of random samples, that is, a sequence of iid random variables drawn from a distribution with mean $\mu$ and variance $\sigma^2 < \infty$.

Then, let the sample avarage be equal to $\bar{X}_n = \frac{X_1+...+X_i+...+X_n}{n}$.

The CLT says, when $n$ becomes large then $\sqrt{n}(\bar{X}_n-\mu)$ is approx. normal with mean $0$ and variance $\sigma^{2}$.

What this theorem is saying is: take a random sample from a population, then compute the sample avarage and keep it aside, then you keep sampling let say for other 100,000 times, and you get additional (possibly different, at least most of them) 100,000 sample avarages. Then, if you look at the sampling distribution of all these sample avarages, it would be approximately normal.

Now, the generic $X_i$ is the sample average obtained from the $i$ random sampling or is simply a random observation obtained from the $i$ random sampling? I mean, is it $\bar{X}_i$ or a single observation? What I'm asking is: should I can consider $X_i$ either as a single observation (in this case each draw means to draw a single value at each trial, and the number of random draws is $n$) or as a sample mean (and in this case I'm drawing samples fixed in size, and in this case $n$ is the number of random samples drawn)?

Does the same interpretation for $X_i$ shall hold when we present the law of large numbers.

Can you tell me if my understanding of the CLT is correct? Maybe it's just a matter of notation.

The classical CLT states:

Let $X_1,...,X_i,...,X_n$ be a sequence of iid random variables drawn from a distribution with mean $\mu$ and variance $\sigma^2 < \infty$.

Then, let the sample avarage be equal to $\bar{X}_n = \frac{X_1+...+X_i+...+X_n}{n}$.

The CLT says, when $n$ becomes large then $\sqrt{n}(\bar{X}_n-\mu)$ is approx. normal with mean $0$ and variance $\sigma^{2}$.

What this theorem is saying is: take a random sample from a population, then compute the sample average and keep it aside, then you keep sampling let's say for additional 100,000 times, and you get additional (possibly different, at least most of them) 100,000 sample averages. Then, if you look at the sampling distribution of all these sample averages, it would be approximately normal.

Now, the generic $X_i$ is the sample average obtained from the $i$ random sampling or is it simply a random observation obtained from the $i$ random sampling? I mean, is it $\bar{X}_i$ or a single observation? What I'm asking is: should I can consider $X_i$ either as a single observation (in this case each draw means to draw a single value at each trial, and the number of random draws is $n$) or as a sample mean (and in this case I'm drawing samples fixed in size, and in this case, $n$ is the number of random samples drawn)?

Shall the same interpretation for $X_i$ hold when we present the law of large numbers?

Can you tell me if my understanding about the CLT is correct? Maybe it's just matter of notation.

The classical CLT states:

Let $X_1,...,X_i,...,X_n$ be a sequence of random samples, that is, a sequence of iid random variables drawn from a distribution with mean $\mu$ and variance $\sigma^2 < \infty$.

Then, let the sample avarage be equal to $\bar{X}_n = \frac{X_1+...+X_i+...+X_n}{n}$.

The CLT says, when $n$ becomes large then $\sqrt{n}(\bar{X}_n-\mu)$ is approx. normal with mean $0$ and variance $\sigma^{2}$.

What this theorem is saying is: take a random sample from a population, then compute the sample avarage and keep it aside, then you keep sampling let say for other 100,000 times, and you get additional (possibly different, at least most of them) 100,000 sample avarages. Then, if you look at the sampling distribution of all these sample avarages, it would be approximately normal.

Now, the generic $X_i$ is the sample average obtained from the $i$ random sampling or is simply a random observation obtained from the $i$ random sampling? I mean, is it $\bar{X}_i$ or a single observation? I think that the same interpretation for $X_i$ shall hold when we present the law of large numbers.

Can you tell me if my understanding about the CLT is correct? Maybe it's just matter of notation.

The classical CLT states:

Let $X_1,...,X_i,...,X_n$ be a sequence of random samples, that is, a sequence of iid random variables drawn from a distribution with mean $\mu$ and variance $\sigma^2 < \infty$.

Then, let the sample avarage be equal to $\bar{X}_n = \frac{X_1+...+X_i+...+X_n}{n}$.

The CLT says, when $n$ becomes large then $\sqrt{n}(\bar{X}_n-\mu)$ is approx. normal with mean $0$ and variance $\sigma^{2}$.

What this theorem is saying is: take a random sample from a population, then compute the sample avarage and keep it aside, then you keep sampling let say for other 100,000 times, and you get additional (possibly different, at least most of them) 100,000 sample avarages. Then, if you look at the sampling distribution of all these sample avarages, it would be approximately normal.

Now, the generic $X_i$ is the sample average obtained from the $i$ random sampling or is simply a random observation obtained from the $i$ random sampling? I mean, is it $\bar{X}_i$ or a single observation? What I'm asking is: should I can consider $X_i$ either as a single observation (in this case each draw means to draw a single value at each trial, and the number of random draws is $n$) or as a sample mean (and in this case I'm drawing samples fixed in size, and in this case $n$ is the number of random samples drawn)?

Does the same interpretation for $X_i$ shall hold when we present the law of large numbers.

Can you tell me if my understanding about the CLT is correct? Maybe it's just matter of notation.

The classical CLT states:

Let $X_1,...,X_i,...,X_n$ be a sequence of random samples, that is, a sequence of iid random variables drawn from a distribution with mean $\mu$ and variance $\sigma^2 < \infty$.

Then, let the sample avarage be equal to $\bar{X}_n = \frac{X_1+...+X_i+...+X_n}{n}$.

The CLT says, when $n$ becomes large then $\sqrt{n}(\bar{X}_n-\mu)$ is approx. normal with mean $0$ and variance $\sigma^{2}$.

What this theorem is saying is: take a random sample from a population, then compute the sample avarage and keep it aside, then you keep sampling let say for other 100,000 times, and you get additional (possibly different, at least most of them) 100,000 sample avarages. Then, if you look at the sampling distribution of all these sample avarages, it would be approximately normal.

Now, the generic $X_i$ is the sample average obtained from the $i$ random sampling or is simply a random observation obtained from the $i$ random sampling? I think that the same interpretation for $X_i$ shall hold when we present the law of large numbers.

Can you tell me if my understanding about the CLT is correct? Maybe it's just matter of notation.

The classical CLT states:

Let $X_1,...,X_i,...,X_n$ be a sequence of random samples, that is, a sequence of iid random variables drawn from a distribution with mean $\mu$ and variance $\sigma^2 < \infty$.

Then, let the sample avarage be equal to $\bar{X}_n = \frac{X_1+...+X_i+...+X_n}{n}$.

The CLT says, when $n$ becomes large then $\sqrt{n}(\bar{X}_n-\mu)$ is approx. normal with mean $0$ and variance $\sigma^{2}$.

What this theorem is saying is: take a random sample from a population, then compute the sample avarage and keep it aside, then you keep sampling let say for other 100,000 times, and you get additional (possibly different, at least most of them) 100,000 sample avarages. Then, if you look at the sampling distribution of all these sample avarages, it would be approximately normal.

Now, the generic $X_i$ is the sample average obtained from the $i$ random sampling or is simply a random observation obtained from the $i$ random sampling? I mean, is it $\bar{X}_i$ or a single observation? I think that the same interpretation for $X_i$ shall hold when we present the law of large numbers.