Solution Found!
Consider the negative binomial distribution given in
Chapter 3, Problem 98E(choose chapter or problem)
Consider the negative binomial distribution given in Definition .
a Show that if \(y \geq r+1\), \(\frac{p(y)}{p(y-1)}-\left(\frac{y-1}{y-r}\right) q\). This establishes a recursive relationship between successive negative binomial probabilities, because \(p(y)=p(y-1) \times\left(\frac{y-1}{y-r}\right) q\).
b Show that \(\frac{p(y)}{p(y-1)}=\left(\frac{y-1}{y-r}\right) q>1\) if \(y<\frac{r-q}{1-q}\). Similarly, \(\frac{p(y)}{p(y-1)}<1\) if \(y>\frac{r-q}{1-q}\).
c Apply the result in part (b) for the case \(r=7\), \(p=.5\) to determine the values of for which \(p(y)>p(y-1)\).
Equation Transcription:
Text Transcription:
y>=r+1
p(y) over p(y-1)-(y-1 over y-r)q
p(y)=p(y-1)x(y-1 over y-r)q
p(y) over p(y-1)=(y-1 over y-r)q>1
y<r-q over 1-q
p(y) over p(y-1)<1
y>r-q over 1-q
r=7
p=.5
p(y)>p(y-1)
Questions & Answers
QUESTION:
Consider the negative binomial distribution given in Definition .
a Show that if \(y \geq r+1\), \(\frac{p(y)}{p(y-1)}-\left(\frac{y-1}{y-r}\right) q\). This establishes a recursive relationship between successive negative binomial probabilities, because \(p(y)=p(y-1) \times\left(\frac{y-1}{y-r}\right) q\).
b Show that \(\frac{p(y)}{p(y-1)}=\left(\frac{y-1}{y-r}\right) q>1\) if \(y<\frac{r-q}{1-q}\). Similarly, \(\frac{p(y)}{p(y-1)}<1\) if \(y>\frac{r-q}{1-q}\).
c Apply the result in part (b) for the case \(r=7\), \(p=.5\) to determine the values of for which \(p(y)>p(y-1)\).
Equation Transcription:
Text Transcription:
y>=r+1
p(y) over p(y-1)-(y-1 over y-r)q
p(y)=p(y-1)x(y-1 over y-r)q
p(y) over p(y-1)=(y-1 over y-r)q>1
y<r-q over 1-q
p(y) over p(y-1)<1
y>r-q over 1-q
r=7
p=.5
p(y)>p(y-1)
ANSWER:
Solution 98E
Step1 of 4:
Let us consider a random variable ‘Y’ it follows negative binomial distribution with parameter
“N, p and r”. If and only if :
Here our goal is:
a). We need to show that if y
this establishes a recursive relationship between successive negative binomial probabilities, because
b). We need to show that similarly
.
c). We need to determine the values of y for which P(y) > P(y-1) when r = 7 and p = 0.5.
Step2 of 4:
a).
Let
Substitute y-1 in place of y then we get