Please note that mathematical proofs require rigorous logic and may take several steps to complete.

Here is a possible proof of the central limit theorem:

Assume we have a sequence of independent and identically distributed random variables, X1, X2, ..., Xn, with mean mu and standard deviation sigma. Let S_n = (X1 + X2 + ... + Xn)/n denote the sample mean of n observations. Then, as n approaches infinity, the distribution of S_n tends to a normal distribution with mean mu and standard deviation sigma/sqrt(n).

Proof:

Moment generating function of S_n: The moment generating function (MGF) of a random variable X is defined as M_X(t) = E(e^(tX)). We can use the MGF to derive the distribution of S_n. Let's find the MGF of S_n:
M_Sn(t) = E(e^(tS_n)) = E(e^(t/n (X1 + X2 + ... + Xn))) = E(e^(t/n X1) * e^(t/n X2) * ... * e^(t/n Xn)) (by independence of X1, X2, ..., Xn) = E(e^(t/n X1))^n (since X1, X2, ..., Xn have the same distribution) = [M_X1(t/n)]^n

Here, we have used the fact that the MGF of a sum of independent random variables is the product of their individual MGFs. Also, the MGF of Xi is simply M_X1(t/n) since they have the same distribution.

Using the MGF to derive the distribution of S_n: Recall that the MGF uniquely determines the distribution of a random variable. Thus, to find the distribution
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