Q. 8.1 It Xhas a variance蟽2, then 蟽, ... [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. 8.1 (page 392)

It $X$ has a variance $蟽^{2}$ , then 蟽, the positive square root of the variance, is called the standard deviation. It $X$ has to mean $渭$ and standard deviation $蟽$ , to show that $P {| X - 渭 | 鈮� k 蟽} 鈮� \frac{1}{k^{2}} .$

Short Answer

Expert verified

Since $| X - 渭 | 鈮� 0$ ,

${(\frac{| X - 渭 |}{k 蟽})}^{2} 鈮� \frac{| X - 渭 |}{k 蟽} 鈮� I_{{| X - 渭 | 鈮� k 蟽}} 鈬� \frac{1}{k^{2}} = E^{未} [{(\frac{| X - 渭 |}{k 蟽})}^{2}] 鈮� E [I_{{| X - 渭 | > k 蟽}}] = P {| X - 渭 | 鈮� k 蟽}$

On the other hand, this result can be proven directly using Chebyshev's inequality. Since $k 蟽 > 0$ ,

$P {| X - 渭 | 鈮� k 蟽} 鈮� \frac{蟽^{2}}{(k 蟽)^{2}} = \frac{1}{k^{2}} .$

Step by step solution

Given Information.

It $X$ has a variance $蟽^{2}$ , then $蟽$ , the positive square root of the variance is called the standard deviation. It $X$ has to mean $渭$ and standard deviation $蟽$ ,

Explanation.

Assume that the random variable $X$ has meant $渭$ and standard deviation $蟽$ , and let $k > 0$ . Then $k 蟽 > 0$ .

Let

$I_{{| X - 渭 | 鈮� k 蟽}} = 1 if | X - 渭 | 鈮� k 蟽 0, otherwise$

Then

$P {| X - 渭 | 鈮� k 蟽} = E [I_{{| X - 渭 | 鈮� k 蟽}}]$

Since $| X - 渭 | > 0$ note that.

${(\frac{| X - 渭 |}{k 蟽})}^{2} 鈮� \frac{| X - 渭 |}{k 蟽} 鈮� I_{{| X - 渭 | > k 蟽}} E [{(\frac{| X - 渭 |}{k 蟽})}^{2}] 鈮� E [I_{{| X - 渭 | 鈮� k 蟽}}] \frac{1}{k^{2} 蟽^{2}} {\overset{铃�}{E [| X - 渭 |^{2}]}}^{= 蟽^{2}} 鈮� P {| X - 渭 | 鈮� k 蟽} 鈩� \frac{1}{k^{2}} 鈮� P {| X - 渭 | 鈮� k 蟽}$

Explanation.

On the other hand, this result can be proven directly using Chebyshev's inequality. Since $k 蟽 > 0$ ,

$P {| X - 渭 | 鈮� k 蟽} 鈮� \frac{蟽^{2}}{(k 蟽)^{2}} = \frac{1}{k^{2}} .$

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91影视

It $X$ has a variance $蟽^{2}$ , then 蟽, the positive square root of the variance, is called the standard deviation. It $X$ has to mean $渭$ and standard deviation $蟽$ , to show that $P {| X - 渭 | 鈮� k 蟽} 鈮� \frac{1}{k^{2}} .$

Short Answer

Step by step solution

Given Information.

Explanation.

Explanation.

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91影视

It Xhas a variance蟽2, then 蟽, the positive square root of the variance, is called the standard deviation. It Xhas to mean 渭and standard deviation蟽, to show thatP{|X-渭|鈮�k蟽}鈮�1k2.

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

Theoretical and Mathematical Physics

Probability and Statistics

Calculus

Geometry

Statistics

Discrete Mathematics

Study anywhere. Anytime. Across all devices.

It $X$ has a variance $蟽^{2}$ , then 蟽, the positive square root of the variance, is called the standard deviation. It $X$ has to mean $渭$ and standard deviation $蟽$ , to show that $P {| X - 渭 | 鈮� k 蟽} 鈮� \frac{1}{k^{2}} .$