Section $1$: Preliminaries

$1$a$)$ Simulating a Multinomial

np$.$ random. multinomial$(n,$ p$\_$array$)$ simulates $n$ draws at random with replacement from the categorical distribution p$\_$array and returns an array of the

observed counts in all the categories.

You have seen a version of this in Data $8.$ The datascience function sample$\_$proportions is just np$.$ random. multinomial with the output converted to

proportions.

Run the cell below to draw $10$ times from a population in which $20\%$ are in Category A$,70\%$ in Category B$,$ and $10\%$ in Category C$.$ The output array contains the

observed counts in Categories A$,$ B$,$ and C$.$

Confirm that it makes sense, by looking at the total count and the count in each category. Then run the cell a few times to see how the observations change.

num$\_$draws $=10$

population$\_$proportions $=[0\mathrm{.}2,0\mathrm{.}7,0\mathrm{.}1]$ # make$\_$array $(0\mathrm{.}2,0\mathrm{.}7,0\mathrm{.}1)$ is also fine

np$.$ randon.multinonial$($num$\_$draws, population$\_$proportions$)$

array$([2,7,1])$

$1$b$)$ Adding Rows to a Table

You know that $t.$ with$\_$columns can be used to add a column or columns to a table $t.$ Sometimes it is useful to grow a table by adding rows. Start by creating the

table with just the column labels and no rows, as follows.

$t=$Table$({[}^{\text{'}}$ Column $A^{\text{'}},$ 'Column $B^{\text{'}},$ 'Colunn $C^{\text{'}}])$

$t$

Column A Column B Column C

a temporary copy of $t$ that you have to name if you want to save it$.$ Run the two cells below and keep track of what happens to $t.$

t$.$append$([1,2,3]$

$t$ has been changed

$t$

Column A Column B Column C

To append another row, use append again.

t$.$append $([4,5,6]$

$1$c$)$ Putting the Two Together

Create a table simulated$\_$counts that has column labels A$,$ B$,$ and C$.$ The table should have two rows. Each row should contain the observed counts in

Categories A$,$ B$,$ and C in $10$ multinomial trials, each of which results in Category A with chance $0\mathrm{.}2,$ Category B with chance $0\mathrm{.}7,$ and Category C with chance $0\mathrm{.}1$ as in

Part a above. The trials in each row should be independent of those in the other row.

You should use no more than $3$ lines of code.

Answer to $1c$

$...$

a$.$

simulated$\_$counts. Please help solve the last part