1 The random variable \(X\) represents the number of cars arriving at a car wash per 10-minute period. From observations over a number of days, an estimate was made of the probability distribution of \(X\). Table 1 shows this estimated probability distribution.
\begin{table}[h]
| \(r\) | 0 | 1 | 2 | 3 | 4 | \(> 4\) |
| \(\mathrm { P } ( X = r )\) | 0.30 | 0.38 | 0.19 | 0.08 | 0.05 | 0 |
\captionsetup{labelformat=empty}
\caption{Table 1}
\end{table}
- In this question you must show detailed reasoning.
Use Table 1 to calculate estimates of each of the following.
- \(\mathrm { E } ( X )\)
- \(\operatorname { Var } ( X )\)
- Explain how your answers to part (a) indicate that a Poisson distribution may be a suitable model for \(X\).
You should now assume that \(X\) can be modelled by a Poisson distribution with mean equal to the value which you calculated in part (a). - Find each of the following.
- \(\mathrm { P } ( X = 2 )\)
- \(\mathrm { P } ( X > 3 )\)
- Given that the probability that there is at least 1 car arriving in a period of \(k\) minutes is at least 0.99 , find the least possible value of \(k\).