---
title: "FW 536 — Day 2 morning lab (Continuous 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 2 morning lab — Continuous distributions problem set

This template matches the **morning** section (Problems 1–4) of
[`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.

Problem 4 loads a real dataset from this Day's `data/` folder:

- `data/sockeye_adult.csv` — daily sockeye counts and flow-corrected qPCR eDNA
  (Levi et al. 2019). Filter to `Sockeyetype == "Sockeye_Adults"`.

Keep the `.Rmd` in this folder so the `read.csv("data/...")` paths resolve.
For setup help see the [R Markdown tutorial](../rmarkdown_tutorial.html).

---

# Setup

```{r}
# Load the packages you'll need. Add or remove as appropriate.
library(ggplot2)
library(dplyr)
library(MASS)        # for glm.nb and negative-binomial helpers
library(effects)
library(emmeans)
```

---

# Problem 1 — Which distribution for a right-skewed body-size sample?

> Body weight (g) from a creel survey: strictly positive, strongly
> right-skewed. Choose a distribution and justify; compute the lognormal
> mean and median for $\log(\text{weight}) \sim \mathcal{N}(5.0, 0.8^2)$;
> explain why a Normal prediction interval would dip below zero.

**My work:**

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

**Answer / biological interpretation:**


---

# Problem 2 — Length–weight allometry and the power law

> Chinook length (cm) vs weight (g). Contrast `lm(weight ~ length)` with
> `lm(log(weight) ~ log(length))`; interpret the log–log slope
> $\hat\beta_1 = 3.05$ as the allometric exponent $b$ in $W = a L^b$;
> give the multiplicative weight effects of a 25% and a doubling length
> change; explain the mean-vs-median back-transform.

**My work:**

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

**Answer / biological interpretation:**


---

# Problem 3 — Line-transect detection as an exponential process

> Exponential detection with rate $\lambda = 0.02$/m. Write the survival
> function $S(x) = e^{-\lambda x}$ and its mean; compute the probability of
> missing an animal at 50 m; find the half-distance $\ln 2/\lambda$; explain
> what memorylessness implies.

**My work:**

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

**Answer / biological interpretation:**


---

# Problem 4 — Are the sockeye counts Poisson? Diagnose overdispersion

> Load `data/sockeye_adult.csv`, subset to `Sockeye_Adults`. Compute the
> mean, variance, and dispersion ratio of `Count`; solve for the NB size $k$
> under $\mathrm{Var} = \mu + \mu^2/k$; simulate Poisson vs NB and compare
> $P(\text{count}=0)$ and $P(\text{count}>400)$; name a mechanism and the
> inferential consequence of using Poisson CIs.

**My work:**

```{r}
# Load and subset the sockeye data
sock <- read.csv("data/sockeye_adult.csv", stringsAsFactors = FALSE)
sock <- subset(sock, Sockeyetype == "Sockeye_Adults")
sock <- sock[!is.na(sock$Count), ]

# mean, variance, dispersion ratio, and NB size k go here
```

**Answer / biological 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. See the
[R Markdown tutorial](../rmarkdown_tutorial.html) for details.)

---

# Submission checklist

- [ ] Answered every morning problem (1–4) from `problem_set.html`.
- [ ] Loaded `data/sockeye_adult.csv` for Problem 4 (`.Rmd` kept in this folder so the path resolves).
- [ ] Translated every log-scale quantity into a multiplicative / biological statement, not just an estimate.
- [ ] Knitted to HTML without errors.
- [ ] Photos (if any) embedded with `self_contained: true` in the YAML.
- [ ] Uploaded the `.html` to the matching Canvas assignment.