Question

1. In your own words, what does the Law of Large Numbers have to say about a coin which is twice as likely to come heads as it is to come up tails? Be sure to give the real world interpretation of any mathematical expressions you use (like x̄n, for example).

2. What does the central limit theorem have to say about that coin from problem 1 (in your own words)? In particular, how does the central limit theorem tell us more than the law of large numbers does?

Please show as much work as possible

thank you

Answer 1

(1)

If the coin is twice as likely to show up heads than tails then the law of large numbers suggest that we toss the coin large number of times (say 10000 times), then we would expect that heads would come up approximately about 2/3rd of the time or in other words 2/3rd of the tosses will result in heads.

The above expectation is based on the assumption that each toss will be under identical conditions, will have the same probability of heads showing up(i.e. 2/3) and there are only two possible outcomes of each toss(heads or tails).

Thus, if Xn denotes the number of heads in n tosses then the law of large number states that when .

(2)

In theory of probabilistic thinking, the central limit theorem (CLT) establishes that, statistically in some situations, when independent random variables are added, their properly normalized sum tends toward a normal distribution even if the original variables themselves are not normally distributed.

Mathematically, if is a random sample under consideration with size n taken from a population with mean μ and finite variance and if is the sample mean, the limiting form of the distribution of as n → ∞ is the standard normal distribution.

In light of the given situation, the CLT states that for large number of coin tosses, the number of times heads come up follows a normal distribution with mean .

The central limit theorem gives us the distribution of the variable under limiting cases where as the Law of Large numbers only give the limiting value.

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