Consider the following question:
A coin has the probability of landing of head equal to 1/4 and is flipped 2000 times.
- Use the law of large numbers, find a lower bound to the probability that the total number of heads lies between 480 and 520.
As far as I know the law of large numbers simply states that for a large number of iid variables, their summation approaches to the expected value of the variable. Does it state anything about its rate of convergence?
The article on law of large numbers in Wikipedia proves it using Chebyshev's Inequality (one of the proofs). I am not sure whether finding the lower bound using Chebyshev's Inequality would count as "using the law of large numbers".
How to solve this problem?
Best Answer
I don't think you can do that. I suppose the question intend to test the Chebyshev's inequality
$$\operatorname{P}( \left| \overline{X}_n-\mu \right| \geq \varepsilon) \leq \frac{\sigma^2}{n\varepsilon^2} $$ . However, the problem is $\mu$ in the formula is sample mean, not the real mean.