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Let X ~ Bin(n, p).a. Show that if x is an integer between

Statistics for Engineers and Scientists | 4th Edition | ISBN: 9780073401331 | Authors: William Navidi ISBN: 9780073401331 38

Solution for problem 28SE Chapter 4

Statistics for Engineers and Scientists | 4th Edition

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Statistics for Engineers and Scientists | 4th Edition | ISBN: 9780073401331 | Authors: William Navidi

Statistics for Engineers and Scientists | 4th Edition

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Problem 28SE

Let X ~ Bin(n, p).

a. Show that if x is an integer between 1 and n inclusive,then

b. Show that if X ~ Bin(n, p), the most probable value for X is the greatest integer less than or equal to np + p. [Hint: Use part (a) to show that P(X = x) ≥ P(X = x - 1) if and only if x ≤ np + p.]

Step-by-Step Solution:

Answer:

Step 1 of 2:

(a)

In this question, we a asked to prove the following.

…………(1)

Where x is an integer between 1 and n inclusive and  is probability.

If  is a random variable whose distribution is binomial with parameter and , then we can write

The probability mass function of a binomial random variable is defined by,

                                                    =    otherwise

                                     ………………..(2)

Replace  with in equation (2), we get

                                             ………………….(3)

 =

(can be written as

can be written as

 simplify the equation

 = 

= (

                                            = (

Hence proved.

Step 2 of 2

Chapter 4, Problem 28SE is Solved
Textbook: Statistics for Engineers and Scientists
Edition: 4
Author: William Navidi
ISBN: 9780073401331

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Let X ~ Bin(n, p).a. Show that if x is an integer between

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