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Problem 45

Suppose that \(X_{i}, i=1,2, \ldots, 20\) are independent random variables, each having a geometric distribution with parameter \(0.8\). Let \(S=X_{1}+\cdots+X_{20}\). Use the central limit theorem \(P(X \geq 18)\).

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The exact numerical answer depends on a custom lookup in the normal distribution table or computation in software.

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Chapter 2

Items are produced in such a manner that the probability of an item being defective is \(p\) (assume unknown). A large number of items say \(n\) are classified as defective or nondefective. How large should \(n\) be so that we may be \(99 \%\) sure that the relative frequency of defective differs from \(p\) by less than \(0.05 ?\)

Chapter 2

Let \(A\) and \(B\) are two independent events. Show that \(A^{c}\) and \(B^{c}\) are also independent events.

Chapter 2

Use CLT to show that
$$
\lim _{n \rightarrow \infty} e^{-n t} \sum_{k=0}^{n} \frac{(n t)^{k}}{k
!}=1=\left\\{\begin{array}{ll}
1, & 0

Chapter 2

Let \(X \sim B(n, p)\). Use CLT to find \(n\) such that $P[X>n / 2] \geq 1-\alpha\(. Calculate the value of \)n\(, when \)\alpha=0.90\( and \)p=0.45$.

Chapter 2

Suppose that diameters of a shaft s manufactured by a certain machine are normal random variables with mean 10 and s.d. \(0.1 .\) If for a given application the shaft must meet the requirement that its diameter falls between \(9.9\) and \(10.2 \mathrm{~cm}\). What proportion of shafts made by this machine will meet the requirement?

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