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

ID: 3178023 • 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

there are some standard statistical distributions like:

normal distribtion

binomial distribution

poisson distribution

when n=100

lets take the example of a businees which sell 3 in ever 4 products every day. lets say he has total 100 products on a day what is the probability that it will sell all of them

according to normal distribution

at 95% confidence interval

z= µ-x/ /n; n=100 in this case

1.96=75-100/sd/100

acording to poisson's distribution

p= (e-) (x) / x!

95%=e-100 (100100) / 100!

95%= e-100 (100100) / 9.33e157

95%=(100100)/9.33e57

95%=e200 /9.33e57

taking log

95=143 -log 9.33

143-2.33

=140.67

thus we can see that there is hugedifference between the confidance interval we take for the same mean value that is 100 in this case. thus even in a larger samle size of 100 it is not approximating to normal distribution.



  

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