Negative Binomial Distribution

1 Rajesh applies once every year for a ticket to a music festival. The probability that he is successful in any particular year is 0.3 , independently of other years.

1. Find the probability that Rajesh is successful for the first time on his 7th attempt.
2. Find the probability that Rajesh is successful for the first time before his 6th attempt.
3. Find the probability that Rajesh is successful for the second time on his 10th attempt.

10 An archer shoots at a target. It may be assumed that each shot is independent of all other shots and that, on average, she hits the bull's-eye with 3 shots in 20 . Find the probability that she requires at least 6 shots to hit the bull's-eye. When she hits the bull's-eye for the third time her total number of shots is \(Y\). Show that $$\mathrm { P } ( Y = r ) = \frac { 1 } { 2 } ( r - 1 ) ( r - 2 ) \left( \frac { 3 } { 20 } \right) ^ { 3 } \left( \frac { 17 } { 20 } \right) ^ { r - 3 } .$$ Simplify \(\frac { \mathrm { P } ( Y = r + 1 ) } { \mathrm { P } ( Y = r ) }\), and hence find the set of values of \(r\) for which \(\mathrm { P } ( Y = r + 1 ) < \mathrm { P } ( Y = r )\). Deduce the most probable value of \(Y\).

2 Independent trials, on each of which the probability of a 'success' is \(p ( 0 < p < 1 )\), are being carried out. The random variable \(X\) counts the number of trials up to and including that on which the first success is obtained. The random variable \(Y\) counts the number of trials up to and including that on which the \(n\)th success is obtained.

1. Write down an expression for \(\mathrm { P } ( X = x )\) for \(x = 1,2 , \ldots\). Show that the probability generating function of \(X\) is $$\mathrm { G } ( t ) = p t ( 1 - q t ) ^ { - 1 }$$ where \(q = 1 - p\), and hence that the mean and variance of \(X\) are $$\mu = \frac { 1 } { p } \quad \text { and } \quad \sigma ^ { 2 } = \frac { q } { p ^ { 2 } }$$ respectively.
2. Explain why the random variable \(Y\) can be written as $$Y = X _ { 1 } + X _ { 2 } + \ldots + X _ { n }$$ where the \(X _ { i }\) are independent random variables each distributed as \(X\). Hence write down the probability generating function, the mean and the variance of \(Y\).
3. State an approximation to the distribution of \(Y\) for large \(n\).
4. The aeroplane used on a certain flight seats 140 passengers. The airline seeks to fill the plane, but its experience is that not all the passengers who buy tickets will turn up for the flight. It uses the random variable \(Y\) to model the situation, with \(p = 0.8\) as the probability that a passenger turns up. Find the probability that it needs to sell at least 160 tickets to get 140 passengers who turn up. Suggest a reason why the model might not be appropriate.