Let X Poisson(). a. Show that if x is a positive integer, then P(X = x) P(X = x 1) = x

Chapter 4, Problem 29

(choose chapter or problem)

Let \(X \sim \text { Poisson }(\lambda)\)

a. Show that if  is a positive integer, then

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

b. Show that if \(X \sim \text { Poisson }(\lambda)\), the most probable value for  is the greatest integer less than or equal to \(\lambda\). [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:

X \sim Poisson (\lambda)

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

X \sim Poisson (\lambda)

\lambda

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

x \leq \lambda