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

Chapter 4, Problem 28SE

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QUESTION:

Let \(X \sim \operatorname{Bin}(n, p)\).

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

$$\frac{P(X=x)}{P(X=x-1)}=\left(\frac{n-x+1}{x}\right)\left(\frac{p}{1-p}\right)$$

b. Show that if \(X \sim \operatorname{Bin}(n, p)\), the most probable value for X is the greatest integer less than or equal to\(n p+p\) . [Hint: Use part (a) to show that \(P(X=x) \geq P(X=x-1)\) if and only if \(x \leq n p+p\).]

Equation Transcription:

.

Text Transcription:

X Tilde Bin(n,p)

P(X=x)/P(X=x-1) = (n-x+1/x) (p/1-p)

np + p

P(X=x) greater than or equal to P (X=x-1)

x less than or equal to np + p

Questions & Answers

QUESTION:

Let \(X \sim \operatorname{Bin}(n, p)\).

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

$$\frac{P(X=x)}{P(X=x-1)}=\left(\frac{n-x+1}{x}\right)\left(\frac{p}{1-p}\right)$$

b. Show that if \(X \sim \operatorname{Bin}(n, p)\), the most probable value for X is the greatest integer less than or equal to\(n p+p\) . [Hint: Use part (a) to show that \(P(X=x) \geq P(X=x-1)\) if and only if \(x \leq n p+p\).]

Equation Transcription:

.

Text Transcription:

X Tilde Bin(n,p)

P(X=x)/P(X=x-1) = (n-x+1/x) (p/1-p)

np + p

P(X=x) greater than or equal to P (X=x-1)

x less than or equal to np + p

ANSWER:

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.


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