Let X ~ Poisson(?).a. Show that if x is a positive

Chapter 4, Problem 29SE

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QUESTION:

Let  Poisson  .

a. Show that if  is a positive integer, then

\(\frac{P(X=x)}{P(Y=x-1)}=\frac{\lambda}{x}\)

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

Equation transcription:

Text transcription:

frac{P(X=x)}{P(Y=x-1)}=frac{\lambda}{x}

P(X=x) geq P(X=x-1)

x leq lambda

Questions & Answers

QUESTION:

Let  Poisson  .

a. Show that if  is a positive integer, then

\(\frac{P(X=x)}{P(Y=x-1)}=\frac{\lambda}{x}\)

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

Equation transcription:

Text transcription:

frac{P(X=x)}{P(Y=x-1)}=frac{\lambda}{x}

P(X=x) geq P(X=x-1)

x leq lambda

ANSWER:

Answer:

Step 1 of 2:

(a)

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

…………(1)

Where x is a positive integer and  is a parameter.

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

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

                                                        =    otherwise

                                     ………………..(2)

Replace  with in equation (2), we get

                                             ………………….(3)

 =

can be written as

 =

 =

Hence proved.


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