Is the Central Limit Theorem Applicable to All Random Variables?

[kristymassi]

i got 2 different answer when i search it..
"The Central Limit Theorem mean of a sampling distribution taken from a single population"
is that true for you guys?

 

[SW VandeCarr]

i got 2 different answer when i search it..
"The Central Limit Theorem holds that the mean of a sampling distribution taken from a single population approaches the actual population mean as the number of samples increases."

is that true for you guys?

That's the definition. Your question was in terms of probabilities. So if a population of surgeons is 30% female, the cumulative mean probability p(f) of repeated random samples of the population will converge to a value p(f)=0.3

 

[wofsy]

i got 2 different answer when i search it..
"The Central Limit Theorem holds that the mean of a sampling distribution taken from a single population approaches the actual population mean as the number of samples increases."

is that true for you guys?

This is the strong law of large numbers not the central limit theorem

 

[SW VandeCarr]

This is the strong law of large numbers not the central limit theorem

Of the choices the OP gave, the CLT is the correct choice, Strictly speaking CTL states that for a sequence of independent identically distributed random variables, each having a finite variance; with increasing numbers (of random variables), their arithmetic mean approaches a normally distributed random variable. The law of large numbers states that this mean will converge to the population mean. In practical terms, the two are quite intertwined when dealing with random sampling from a defined static population.

 

[wofsy]

Of the choices the OP gave, the CLT is the correct choice, Strictly speaking CTL states that for a sequence of independent identically distributed random variables, each having a finite variance; with increasing numbers (of random variables), their arithmetic mean approaches a normally distributed random variable. The law of large numbers states that this mean will converge to the population mean. In practical terms, the two are quite intertwined when dealing with random sampling from a defined static population.

The Central Limit Theorem says much more to me than just the convergence of means - and it requires finite variance, a restriction that is not need for the strong law of large numbers.

 

[mathman]

The essence of the central limit theorem is that a sum of random variables (number increasing without limit), under certain conditions and properly normalized, will have a distribution approaching the normal distribution.