Let X Bin(n, p). a. Show that if x is an integer between 1 and n inclusive, then P(X =

Chapter 4, Problem 28

(choose chapter or problem)

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

a. Show that if  is an integer between \(1 \text { and } n\) 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  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 \sim Bin n, p)

1 and  n

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

X \sim Bin (n, p)

n p+p

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

x \leq n p+p