Q. 8.6 8.6 . In Self-Test Problem 8.5, ... [FREE SOLUTION]

91影视

A First Course in Probability

Sheldon M. Ross

$Math Studyset 91影视 Explanations$ Math

9th Edition 路 432 pages

ISBN: 9780321794772

Chapter 8: Q. 8.6 (page 393)

8.6 . In Self-Test Problem $8.5$ , how many components would one need to have on hand to be approximately $90$ percent certain that the stock would last at least $35$ days?

Short Answer

Expert verified

The components is $n = 56$ .

Step by step solution

Given information

The components need to have on hand, approximately $90$ percent. The stock would last at least $35$ days.

Explanation

Let $X_{i}$ denote the lifetime of $i$ th component. Understand that $X_{1}, X_{2}, 鈥�$ is a sequence of independent and identically distributed random variables, each having a mean.
$渭 = E [X_{i}] = {鈭�}_{0}^{1} x f (x) d x = {鈭�}_{0}^{1} 2 x^{2} d x = {2 \frac{x^{3}}{3}|}_{0}^{1} = \frac{2}{3}$
Then the variance is,
$蟽^{2} = Var (X_{i}) = {鈭�}_{0}^{1} x^{2} f (x) d x - {(\frac{2}{3})}^{2} = {鈭�}_{0}^{1} 2 x^{3} d x - \frac{4}{9} = {2 \frac{x^{4}}{4}|}_{0}^{1} - \frac{4}{9} = \frac{1}{18}$
Let $S_{n} = {鈭�}_{i = 1}^{n} X_{i}$ indicates the time of $n$ th failure.

Verify that the value of $n$ is satisfied.
$P \{S_{n} 鈮� 35\} 鈮� . 90 ?$

Explanation

Using the central limit theorem:

$.90 鈮� P \{S_{n} 鈮� 35\} = 1 鈭� P \{S_{n} 鈮� 35\} = 1 鈭� P \{\frac{S_{n} 鈭� n 渭}{蟽 \sqrt{n}} 鈮� \frac{35 鈭� n 渭}{蟽 \sqrt{n}}\} 鈮� 鈮� 1 - 桅 (\frac{35 - n 渭}{蟽 \sqrt{n}}) 鈬� 桅 (\frac{35 - n 渭}{蟽 \sqrt{n}}) 鈮� . 10 鈬� \frac{35 - n 渭}{蟽 \sqrt{n}} 鈮� 桅^{- 1} (. 10) = - 1.28$

Then,

$\begin{matrix} \frac{\sqrt{18} (105 鈭� 2 n)}{3 \sqrt{n}} 鈮� 鈭� 1.28 \end{matrix} n = 56$

Hence, the components $n = 56$

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91影视

8.6 . In Self-Test Problem $8.5$ , how many components would one need to have on hand to be approximately $90$ percent certain that the stock would last at least $35$ days?

Short Answer

Step by step solution

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Explanation

Explanation

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91影视

8.6 . In Self-Test Problem 8.5, how many components would one need to have on hand to be approximately 90percent certain that the stock would last at least 35days?

Short Answer

Step by step solution

Given information

Explanation

Explanation

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Probability and Statistics

Calculus

Discrete Mathematics

Applied Mathematics

Geometry

Logic and Functions

Study anywhere. Anytime. Across all devices.

8.6 . In Self-Test Problem $8.5$ , how many components would one need to have on hand to be approximately $90$ percent certain that the stock would last at least $35$ days?