---
title: "FW 536 — Day 1 afternoon lab (Discrete distributions) — Your Name"
author: "Your Name Here"
date: "`r Sys.Date()`"
output:
  html_document:
    self_contained: true
    toc: true
    toc_depth: 3
    toc_float: true
    code_folding: show
    theme: cosmo
---

```{r setup, include=FALSE}
knitr::opts_chunk$set(echo = TRUE, message = FALSE, warning = FALSE)
```

# Day 1 afternoon lab — Discrete distributions problem set

This template matches the afternoon problems (7–12) in [`problem_set.html`](problem_set.html).
Read each problem statement there, write your work below, **Knit** to HTML, and
upload the `.html` to the matching Canvas assignment.

**Data:** Problem 9 uses `data/coyote_scat_counts.csv`. Keep the `data/` folder
alongside this `.Rmd` file so the `read.csv()` call resolves.

For setup help see the [R Markdown tutorial](../rmarkdown_tutorial.html).

---

# Setup

```{r}
# Load packages you'll need. Add or remove as appropriate.
library(ggplot2)
library(dplyr)
```

---

# Problem 7 — Sea turtle nest success

> n = 18 nests, each hatching successfully with p = 0.62, independently.
> (a) Model + mean and variance. (b) P(exactly 12). (c) P(at least 15).
> (d) What p gives at least an 80% chance of >= 15 successes? Find it numerically.

**My work:**

```{r}
# Your R code here
```

**Answer / interpretation:**


---

# Problem 8 — Wolf approaches at an active den

> Wolf approaches occur at rate lambda = 1.8 per 12-h period.
> (a) Model + independence assumption. (b) P(0 in 12 h). (c) P(more than 3 in 12 h).
> (d) P(exactly 5 in 36 h) — scale lambda with the window.

**My work:**

```{r}
# Your R code here
```

**Answer / interpretation:**


---

# Problem 9 — Coyote scat counts: is Poisson enough?

> Load `data/coyote_scat_counts.csv` (120 grid cells, column `scats`).
> (a) Sample mean and variance — what do you notice about mean vs variance?
> (b) Method-of-moments NB dispersion k = mu^2 / (var - mu).
> (c) Compare observed P(Y=0) and P(Y>15) against Poisson(mu) and NB(mu, k).
> (d) Histogram with Poisson and NB pmfs overlaid; name a mechanism for the over-dispersion.

**My work:**

```{r}
coyote <- read.csv("data/coyote_scat_counts.csv")
str(coyote)
mu <- mean(coyote$scats)
v  <- var(coyote$scats)
c(mean = mu, variance = v)
# Your fitted k, model comparison, and overlay plot here
```

**Answer / interpretation:**


---

# Problem 10 — Owl pellet contents

> Prey composition voles 0.65, shrews 0.20, mice 0.10, birds 0.05; 15 items per pellet.
> (a) P(exact composition (10, 3, 1, 1)). (b) Marginal distribution of shrews (name, mean,
> variance). (c) Expected total mice across 50 pellets. (d) Simulate 5,000 pellets and
> confirm a category's marginal matches the binomial from (b).

**My work:**

```{r}
set.seed(1)
# Your R code here
```

**Answer / interpretation:**


---

# Problem 11 — Pooling counts across sites creates over-dispersion

> Poisson counts with lambda_A = 2, lambda_B = 8; 50 mussels from each site, pooled.
> (a) Naive pooled mean/variance — still Poisson? (b) Simulate and show over-dispersion.
> (c) Derive the exact mean and variance via the law of total variance and compare.
> (d) Why are real count data over-dispersed even when Poisson within each unit?

**My work:**

```{r}
set.seed(1)
# Your R code here
```

**Answer / interpretation:**


---

# Problem 12 — Disease surveillance sample size

> Prevalence p = 0.15; want at least 95% probability of detecting >= 1 positive.
> (a) Write P(at least one positive) and the 95% inequality. (b) Smallest n (log form +
> R check). (c) Repeat for prevalence 0.02 and comment on how effort scales.

**My work:**

```{r}
# Your R code here
```

**Answer / interpretation:**


---

# (Optional) Embed a photo of hand-written work

For problems easier to do on paper, photograph and embed:

```{r, echo=FALSE, eval=FALSE, out.width="60%", fig.cap="Hand-derivation for Problem N"}
knitr::include_graphics("images/problemN_derivation.jpg")
```

(Change `eval=FALSE` to `eval=TRUE` after saving an image in an `images/` folder next
to this `.Rmd`. See the [R Markdown tutorial](../rmarkdown_tutorial.html) for details.)

---

# Submission checklist

- [ ] Answered every afternoon problem (7–12) from `problem_set.html`.
- [ ] Loaded `data/coyote_scat_counts.csv` for Problem 9 (kept the `data/` folder next to this file).
- [ ] Set a seed before each simulation (Problems 10 and 11).
- [ ] Knitted to HTML without errors.
- [ ] Photos (if any) embedded with `self_contained: true` in the YAML.
- [ ] Uploaded the `.html` to the matching Canvas assignment.