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

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

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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:

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

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

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