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Let X_1, X_2, ...be a sequence of independent and identically distributed random

ID: 3177786 • Letter: L

Question

Let X_1, X_2, ...be a sequence of independent and identically distributed random variables with distribution F, having a finite mean and variance. Whereas the central limit theorem states that the distribution of Sigma^n_i = 1 X_i approaches a normal distribution as n goes to infinity, it gives us no information about how large n need be before the normal becomes a good approximation. Whereas in most applications, the approximation yields good results whenever n greaterthanorequalto 20, and oftentimes for much smaller values of n, how large a value of n is needed depends on the distribution of X_i. Give an example of a distribution F such that the distribution of Sigma^100_i = 1 X_i is not close to a normal distribution.

Explanation / Answer

For Xi belonging to poisson distribution with mean lambda, summation of Xi from 1 to 100 follows a poisson distribution with mean 100*lamda

This does not mean that this distribution violates Central limit theorerm, but it requires larger values of n for it to hold

So even if n=100 it follows more of poisson than normal distribition as n tending to infinity.

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Let X_1, X_2, ..., X_n, and Y_1, Y_2, ..., Y_m be independent random variables,
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