## Exercises

### Exercise 1

```{exercise}

:label: exercise_1

Recall that $n!$ is read as "$n$ factorial" and defined as

$n! = n \times (n - 1) \times \cdots \times 2 \times 1$.

In particular, write a function `factorial` such that `factorial(n)` returns $n!$

for any positive integer $n$.

```

### Exercise 2

```{exercise}

:label: exercise_2

The [binomial random variable](https://en.wikipedia.org/wiki/Binomial_distribution) $Y \sim Bin(n, p)$ represents the number of successes in $n$ binary trials, where each trial succeeds with probability $p$.

Without any import besides `from numpy.random import uniform`, write a function

`binomial_rv` such that `binomial_rv(n, p)` generates one draw of $Y$.

Hint: If $U$ is uniform on $(0, 1)$ and $p \in (0,1)$, then the expression `U < p` evaluates to `True` with probability $p$.

```

### Exercise 3

```{exercise}

:label: exercise_3

First, write a function that returns one realization of the following random device

- If a head occurs `k` or more times within this sequence, pay one dollar.

Use no import besides `from numpy.random import uniform`.

```

## Solutions

### Exercise 1

```{solution-start} exercise_1

:label: solution_1

:class: dropdown

Here's one solution.

```

```{code-cell} python3

def factorial(n):

factorial(4)

```

### Exercise 2

```{solution-end}

```

```{solution-start} exercise_2

:label: solution_2

:class: dropdown

````

```{code-cell} python3

from numpy.random import uniform

binomial_rv(10, 0.5)

```

### Exercise 3

```{solution-end}

```

```{solution-start} exercise_3

:label: solution_3

:class: dropdown

Here's a function for the first random device.

```

```{code-cell} python3

from numpy.random import uniform

draw_new(3)

```

```{solution-end}

```