Concept explainers
Refer to Chebyshev’s inequality given in Exercise 44. Calculate P(|X - μ| ≥ kσ) for k = 2 and k = 3 when X ∼ Bin(20, .5), and compare to the corresponding upper bound. Repeat for X ∼ Bin(20, .75).
Find the value of
Compare the upper bound for the different k value.
Answer to Problem 67E
For
for
In each cases, the values of
Explanation of Solution
Given info:
The Chebyshev’s inequality states that for any probability distribution of an rv X and any number k that is at least 1, then
Calculation:
The value of
The expectation and standard deviation is obtained as given below:
For k = 2
Where,
Procedure for binomial distribution table value:
From the table A.1 of Cumulative Binomial probabilities,
- Locate n=20
- Along with n=20, choose x=5,14
- Then, obtain the table value corresponding to p=0.5.
The value of
Hence,
For k=3
Where,
Procedure for binomial distribution table value:
From the table A.1 of Cumulative Binomial probabilities,
- Locate n=20
- Along with n=20, choose x=3,16
- Then, obtain the table value corresponding to p=0.5.
The value of
Hence,
Hence, for
For
The expectation and standard deviation:
For k=2
Where,
Procedure for binomial distribution table value:
From the table A.1 of Cumulative Binomial probabilities,
- Locate n=20
- Along with n=20, choose x=11,18
- Then, obtain the table value corresponding to p=0.75.
The value of
Hence,
For k=3
Where,
Procedure for binomial distribution table value:
From the table A.1 of Cumulative Binomial probabilities,
- Locate n=20
- Along with n=20, choose x=9
- Then, obtain the table value corresponding to p=0.75.
The value of
Hence,
Thus, for
The Chebyshev’s bound
For k = 2,
For k = 3,
In each cases, the values of
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Chapter 3 Solutions
Probability and Statistics for Engineering and the Sciences
- Algebra & Trigonometry with Analytic GeometryAlgebraISBN:9781133382119Author:SwokowskiPublisher:CengageCalculus For The Life SciencesCalculusISBN:9780321964038Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.Publisher:Pearson Addison Wesley,