Solution Found!
If Y is a negative binomial random variable, define Y ? =
Chapter 3, Problem 100E(choose chapter or problem)
If is a negative binomial random variable, define \(Y^{*}=Y-r\). If
is interpreted as the number of the trial on which the
th success occurs, then \(Y^{*}\) can be interpreted as the number of failures before the
th success.
a If \(Y^{*}=Y-r\), \(P\left(Y^{*}=y\right)=P(Y-r=y)=P(Y=y+r)\) for \(y=0\), 1, 2,..., show that \(P\left(Y^{*}=y\right)=\left(\begin{array}{c}
y+r-1 \\
r-1
\end{array}\right) p^{r} q^{y}
\), \(y=0\), 1, 2…..
b Derive the mean and variance of the random variable \(Y^{*}\) by using the relationship \(Y^{*}=Y-r\), where is negative binomial and the result in Exercise
Equation Transcription:
Text Transcription:
Y*=Y-r
Y*
Y*=Y-r
P(Y*=y)=P(Y-r=y)=P(Y=y+r)
y=0
P(Y*=y)=( _r-1^y+r-1)p^r q^y
y=0
Y*
Y*=Y-r
Questions & Answers
QUESTION:
If is a negative binomial random variable, define \(Y^{*}=Y-r\). If
is interpreted as the number of the trial on which the
th success occurs, then \(Y^{*}\) can be interpreted as the number of failures before the
th success.
a If \(Y^{*}=Y-r\), \(P\left(Y^{*}=y\right)=P(Y-r=y)=P(Y=y+r)\) for \(y=0\), 1, 2,..., show that \(P\left(Y^{*}=y\right)=\left(\begin{array}{c}
y+r-1 \\
r-1
\end{array}\right) p^{r} q^{y}
\), \(y=0\), 1, 2…..
b Derive the mean and variance of the random variable \(Y^{*}\) by using the relationship \(Y^{*}=Y-r\), where is negative binomial and the result in Exercise
Equation Transcription:
Text Transcription:
Y*=Y-r
Y*
Y*=Y-r
P(Y*=y)=P(Y-r=y)=P(Y=y+r)
y=0
P(Y*=y)=( _r-1^y+r-1)p^r q^y
y=0
Y*
Y*=Y-r
ANSWER:
Solution 100E
Step1 of 3:
Let us consider a random variable ‘Y’ it follows negative binomial distribution with parameter “N, p and r”. DefineIf Y is interpreted as the number of the trial on which the r th success occurs, then
can be interpreted as the number of failures before the r th success.
Here our goal is:
a). If P(
= y) = P(Y - r = y)
For y = 0,1,2,..., show that P(= y)
b). We need to derive mean and variance of random variable ‘Y’ by using the relationship
Step2 of 3:
a).
We have P(