Riemann surface
matlab

Example 1.1.3 Suppose that in a horse race there are eight horses. If you correctly predict which horse will win the race and which horse will come in second and wager to that effect, you are said to “win the exacta”.

Win the exacta: Need to purchase $(8)(7)=56$ betting tickets.

Outcomes of all eight positions: $8 !=40320$ different ways.

Rule 1.1.3 (Multinomial coefficient) If a group of $n$ objects is composed of $n_1$ objects of type $1, n_2$ identical objects of type $2, \ldots, n_r$, identical objects of type $r$, the number of distinguishable arrangements into a row, denoted by
$$\left(\begin{array}{c} n \ n_1, \ldots, n_r \end{array}\right)=\frac{n !}{n_{1} ! \ldots n_{r} !} .$$
In particular, $\left(\begin{array}{l}n \ k\end{array}\right)=\frac{n !}{k !(n-k) !}$ if $n_1=k$ and $n_2=n-k$.

Example 1.1.4 (In example 2) Suppose $A$ and $B$ are identical. We will denote them by the letter $X$, then

Solution
The $t$ test may be used if the population is normal or if the sample size is at least 30 . The normal probability plot shows two data values outside the bounds, indicating that the data are not normally distributed. Also, the sample of size $n=8$ is not at least 30 . Therefore, the conditions for performing the $t$ test for the population mean are not met. (The unusual data value of 1016 hurricane-related deaths for 2005 is the result of Hurricanes Katrina and Rita.)

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