Let X1, X2, ... be a sample from a population that is geometrically distributed with p = }. (a) Use a suitable version of the Central Limit Theorem to estimate the probability 800 PEx; > 2450 (b) The negative binomial random variable was defined as a sum of a sample of geometric random variables. Why might this approach be preferable to just using the negative binomial PMF?

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**Geometric Distribution and the Central Limit Theorem**

Let \( X_1, X_2, \ldots \) be a sample from a population that is geometrically distributed with \( p = \frac{1}{4} \).

**(a)** Use a suitable version of the Central Limit Theorem to estimate the probability

\[
P\left(\sum_{i=1}^{800} X_i \geq 2450\right).
\]

**(b)** The negative binomial random variable was defined as a sum of a sample of geometric random variables. Why might this approach be preferable to just using the negative binomial PMF? 

This problem explores the use of the Central Limit Theorem (CLT) for approximating probabilities involving sums of geometric random variables and discusses the benefits of defining a negative binomial random variable in this manner.
Transcribed Image Text:**Geometric Distribution and the Central Limit Theorem** Let \( X_1, X_2, \ldots \) be a sample from a population that is geometrically distributed with \( p = \frac{1}{4} \). **(a)** Use a suitable version of the Central Limit Theorem to estimate the probability \[ P\left(\sum_{i=1}^{800} X_i \geq 2450\right). \] **(b)** The negative binomial random variable was defined as a sum of a sample of geometric random variables. Why might this approach be preferable to just using the negative binomial PMF? This problem explores the use of the Central Limit Theorem (CLT) for approximating probabilities involving sums of geometric random variables and discusses the benefits of defining a negative binomial random variable in this manner.
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