10 The discrete random variables \(X\) and \(Y\) have distributions as follows: \(X \sim \mathrm {~B} ( 20,0.3 )\) and \(Y \sim \operatorname { Po } ( 3 )\).
The spreadsheet in Fig. 10 shows a simulation of the distributions of \(X\) and \(Y\). Each of the 20 rows below the heading row consists of a value of \(X\), a value of \(Y\), and the value of \(X - 2 Y\).
\begin{table}[h]
| 1 | A | B | C |
| 1 | X | Y | \(X - 2 Y\) |
| 2 | 6 | 6 | -6 |
| 3 | 5 | 4 | -3 |
| 4 | 8 | 1 | 6 |
| 5 | 6 | 5 | -4 |
| 6 | 6 | 3 | 0 |
| 7 | 8 | 1 | 6 |
| 8 | 6 | 4 | -2 |
| 9 | 5 | 4 | -3 |
| 10 | 7 | 4 | -1 |
| 11 | 8 | 3 | 2 |
| 12 | 6 | 2 | 2 |
| 13 | 5 | 1 | 3 |
| 14 | 6 | 1 | 4 |
| 15 | 5 | 4 | -3 |
| 16 | 7 | 2 | 3 |
| 17 | 5 | 2 | 1 |
| 18 | 4 | 4 | -4 |
| 19 | 5 | 0 | 5 |
| 20 | 5 | 1 | 3 |
| 21 | 4 | 2 | 0 |
| nn | | | |
\captionsetup{labelformat=empty}
\caption{Fig. 10}
\end{table}
- Use the spreadsheet to estimate each of the following.
The mean of 50 values of \(X - 2 Y\) is denoted by the random variable \(W\).
- Calculate an estimate of \(\mathrm { P } ( W > 1 )\).