Solution Found!
Let X ~ Poisson(?).a. Show that if x is a positive
Chapter 4, Problem 29SE(choose chapter or problem)
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.