Solved : Central limit theorem query

Hey there,

I’m a bit confused and want to clarify with you before posting an actual question.

Can you tell me that in CLT, we have X= X1+X2+…+Xn

What is this ‘n’ here? Is it the size of the sample or is it number of samples. It’s really confusing because both of the above stated meanings are being used in the videos.

Acc. To the theorem, it says no. X1, X2, … , Xn are random samples. But then sir calls it the sample size.

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Thanks to @sanjayk for answer.

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I agree, things seems a bit confusing at times.

See if this description helps.

Think of X_i~'s (i.e. X_1, X_2…etc) ONLY as Random Variables. For e.g. number appearing on a single throw of a dice, or sum of numbers appearing on 5 throws of a dice, or mean of numbers appearing on 3 throws of a dice).
Point is, you have control over defining the experiment and what corresponding R.V. is associated with experiment outcome. Experiment can be simple or complex, single roll or multiple roll, observation/R.V. can be number showing up on the dice or some sum or mean of numbers showing up as a result of multiple rolls etc etc.

Next, you do that experiment n number of times, so that you have observation of n R.V.s i.e. X_1, X_2,...,X_n.
CLT says, if you define another R.V. X such that, its the sum of X_i 's:

X=X_1+X_2+...+X_n

then distribution of X will be normal as n\to\infty.
Recall, what distribution of X means. To visualize:

  1. Do the experiment and find X_i (for i=1 till n)
  2. next, calculate X i…e the sum of X_i's
  3. Perform step 1 and step 2 again and again to have several values of X and then plot the values of X. The plot will resemble a normal distribution.

I hope the above description helped you understand what is n.

R.V. X_i can resemble a sample statistic, for e.g. you draw a sample and calculate the sample mean i.e. (X_i). Note that sample size used for calculating sample mean (X_i) here is different from the n used to calculate the R.V X. Right? (n here will be number of samples you will take, such that each will have a sample mean, which you will use for calculating X)

X_i can be also be a R.V. coming from a single roll of a dice and be the number showing up. In this case, you can simply take n observations and use it to calculate the X, the r.v. representing the sum. Here, you might be tempted to say, you draw a sample of size n and its sum is X.

Did I leave you more confused? :slight_smile:
As long as you can remember (1)X_i are independent r.v.s, and (2)the sum of n such R.V.s will be another R.V. (3) which follows a normal distribution as (4) n\to\infty, you are good.

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