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Central Limit Theorem - Exercise set 1

This exercise set contains some solved exercises on the Central Limit Theorem. The theory needed to solve these exercises is introduced in the lecture entitled Central Limit Theorem.

Exercise 1.1

Let [eq1] be a sequence of independent Bernoulli random variables with parameter $p= rac{1}{2}$, i.e. a generic term X_n of the sequence has support[eq2]and probability mass function:[eq3]

Use a Central Limit Theorem to derive an approximate distribution for the mean of the first $100$ terms of the sequence.

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The sequence [eq1] is and IID sequence. The mean of a generic term of the sequence is:[eq5]The variance of a generic term of the sequence can be derived thanks to the usual formula for computing the variance ([eq6]):[eq7]Therefore, the sequence [eq1] satisfies the conditions of Lindeberg-Lévy Central Limit Theorem (IID, finite mean, finite variance). The mean of the first $100$ terms of the sequence is:[eq9]Using the Central Limit Theorem to approximate its distribution, we obtain:[eq10]or[eq11]

Exercise 1.2

Let [eq1] be a sequence of independent Bernoulli random variables with parameter $p= rac{1}{2}$, as in the previous exercise. Let [eq13] be another sequence of random variables such that[eq14]

Suppose [eq15] satisfies the conditions of a Central Limit Theorem for correlated sequences. Derive an approximate distribution for the mean of the first n terms of the sequence [eq15].

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The sequence [eq1] is and IID sequence. The mean of a generic term of the sequence is:[eq18]The variance of a generic term of the sequence is:[eq19]The covariance between two successive terms of the sequence is:[eq20]The covariance between two terms that are not adjacent ($Y_{n}$ and $Y_{n+j}$, with $j>1$) is:[eq21]The long-run variance is:[eq22]The mean of the first n terms of the sequence [eq15] is:[eq24]Using the Central Limit Theorem for correlated sequences to approximate its distribution, we obtain:[eq25]or[eq26]

Exercise 1.3

Let Y be a binomial random variable with parameters $n=100$ and $p= rac{1}{2}$ (you need to read the lecture entitled Binomial distribution in order to be able to solve this exercise). Using the Central Limit Theorem, show that a normal random variable X with mean $mu =50$ and variance $sigma ^{2}=25$ can be used as an approximation of Y.

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A binomial random variable Y with parameters $n=100$ and $p= rac{1}{2}$ can be written as:[eq27]where X_1, ..., $X_{100}$ are mutually independent Bernoulli random variables with parameter $p= rac{1}{2}$. Thus:[eq28]In the first exercise, we have shown that the distribution of $overline{X}_{100}$ can be approximated by a normal distribution:[eq29]Therefore, the distribution of Y can be approximated by:[eq30]Thus, Y can be approximated by a normal distribution with mean $mu =50$ and variance $sigma ^{2}=25$.

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