---
title: "STAT 331 Week 8 Day 2 Handout"
format: html
embed-resources: true
---

```{r}
#| label: setup

library(tidyverse)
```

## Statistical Distributions

For the following distributions:

1. What is the nickname for the distribution in R?
2. Draw or describe what the distribution looks like
3. What parameter(s) define the distribution? 
4. What are those parameter(s) called in the R functions affiliated with that distribution? What are the default values for the parameter(s) in the R functions?


- Uniform



- Normal 



- t
   
   
   
-  Chi-square 



-  Binomial

  + Also, how is the binomial distribution different from other distributions?


## Distribution Functions in R


-   `r` stands for:

    -   What is the required argument to the `r` functions?

    -   What does the `rnorm()` function return?

-   `p` stands for:

    -   What is the required argument to the `p` functions?

    -   What does the `pchisq()` function return?
    
-   `q` stands for:

    -   What is the required argument to the `q` functions?

    -   What does the `qt()` function return?

-   `d` stands for:

    -   What is the required argument to the `d` functions?

    -   What does the `dbinom()` function return?

## Random Samples

1. I want to randomly permute the elements of a vector `1:10`. How can I use the `sample()` function to do this? How do I ensure the permutation is the same every time this code is run?

```{r}


```

2. Emulate randomly selecting a bingo number (a number between 1 and 75).

```{r}


```

3. What happens if you try to sample more items from a vector (`size`) than the length of the vector? Why?

```{r}
sample(1:10, size = 20)
```

4. Take a random selection of 10 songs from the `billboard` data. (This data is built into R.) 

```{r}
billboard 
```

## Iterating Simulations

Birthday Problem: Simulate the approximate probability that at least two people have the same birthday (same day of the year, not considering year of birth or leap years).

1. Write a function to ...

+ simulate the birthdays of `n` random people (assuming it is equally likely to be born any day of the year).
+ count how many birthdays are shared.
+ return whether or not a shared birthday exists.

```{r}
bday_prob <- function( ){
  
  
  
}


```


2. Now use a `map()` function to repeat this simulation 1,000 times.

```{r}

sim_results <- map_lgl(.x = ,
                       .y = ~ bday_prob(n = 50))

```

3. What proportion of the time did at least two people have the same birthday?


```{r}

```


4. What does the `r ...` syntax below do? What do we call this?

`r mean(sim_results)*100`% of the iterations contain at least two people with the same birthday.

## Simulating Multiple Datasets

1. What does the function `sim_ht()` do?

```{r}
sim_ht <- function(n = 200, avg, std){
  
  tibble(person = 1:n,
         ht = rnorm(n = n, mean = avg, sd = std))
  
}
```

2. What does the `crossing()` function do? What is the purpose of using it here?

```{r}
parameters <- crossing(mean_ht = seq(from = 60, to = 78, by = 6),
                       std_ht  = c(3, 6))
```

3. What are the arguments to the `pmap` family of functions?



4. In the code below...

a. Where do the variables `mean_ht` and `std_ht` come from?
b. Why do we name the elements of the list `avg` and `std`?
c. Why is the `ht_data` column so weird!? What is it's data structure?

```{r}
parameters |> 
   mutate(ht_data = pmap(.l = list(avg = mean_ht, 
                                   std = std_ht), 
                         .f = sim_ht)
          )
```

5. What does the `unnest` function do?

```{r}
sim_dat <- parameters |> 
   mutate(ht_data = pmap(.l = list(avg = mean_ht, 
                                   std = std_ht), 
                         .f = sim_ht)
          ) |> 
  unnest(cols = ht_data)

sim_dat
```

6. Where are the `person` and `ht` columns coming from?


7. How many rows should I have for each `mean_ht`, `std_ht` combination?


8. Calculate empirical (simulated) mean and standard deviation of the generated heights for each simulated dataset. Do they align with our expectation?


```{r}

```