Apply Chebychev’s inequality to: (i) arrive at ‘Chebychev’s bounds’ for the sum of ‘n’ Bernoulli variables; and (ii) state the Law of Large Numbers (LLN) in general.

I)

Chebychev's inequality is a useful tool in probability theory that gives an upper bound on the probability that a random variable deviates significantly from its mean. The inequality states that, for a given random variable X with mean μ and standard deviation σ, the probability that X deviates from μ by more than kσ is at most 1/k^2.

Chebychev's inequality can be applied to the sum of 'n' Bernoulli variables, which are random variables that take the values 0 or 1 with some given probability p. If X1, X2, X3, ..., Xn are n Bernoulli variables with the same probability of success p, then the sum S = X1 + X2 + X3 + ... + Xn is a binomial random variable with mean n * p and standard deviation √(n * p * (1-p)).

Using Chebychev's inequality, we can find the bounds for the sum of n Bernoulli variables, also known as Chebychev's bounds.

The bounds are given by:

n * p - k * √(n * p * (1-p)) <= S <= n * p + k * √(n * p * (1-p))

where k is a constant that determines the deviation from the mean.

If we set k = 1, the Chebychev's bounds become:

n * p - √(n * p * (1-p)) <= S <= n * p + √(n * p * (1-p))

This means that with a probability of at least (1 - 1/k^2) = 1 - 1/1^2 = 1 - 1 = 0.75, the sum of the n Bernoulli variables will be between the bounds.

If we set k = 2, the Chebychev's bounds become:

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