What we did today, in one paragraph
Plain-language summary
We started with the question:
"What does it mean to give a number to how likely something is?" We built up the rules for how those numbers behave (probability axioms), how to combine them when one thing depends on another (conditional probability), and how to flip them around (Bayes' rule). Then we introduced
random variables — numerical summaries of random outcomes — and met five workhorse "named" distributions you'll see all week:
Bernoulli, Binomial, Poisson, Negative Binomial, and Multinomial. The lab problems ground each of these in real wildlife and ecology examples.
The big-picture story of the whole course
By the end of the week, you'll be fitting models like glm(y ~ x, family=binomial) or writing Bayesian models in Nimble. Underneath every such command is the same idea:
Key idea
Every statistical model is a probability story about how your data were made. You picked it. The computer's job is to find parameter values that make that story most plausible given what you saw. Your job — the part that doesn't go away no matter how fancy the tools get — is to pick a story that resembles how the data really were generated.
That's why we spend Day 1 on probability and distributions, not on R syntax. If you know the right story, the R will follow.
What is a probability?
A probability is a number between 0 and 1 that summarizes how often something would happen if we could replay reality many times. There are two ways people interpret it:
- Frequentist: "If I drew this sample 10,000 times, the proportion of times I'd see X is the probability of X."
- Bayesian: "Given everything I currently know, my degree of belief that X is true is..."
Both interpretations follow the same algebra. We'll use frequentist language all week until Day 4, when we'll switch to Bayesian and you'll see why the second interpretation buys you something extra.
Three rules you must internalize
Plain-language summary
Multiplication rule (AND). If two events are independent, the probability they both happen is the product of their individual probabilities.
Ecology example: A camera has 0.03 chance of detecting a marten on any night. Two consecutive nights: $0.03 \times 0.03 = 0.0009$ chance of detection on both, if the nights are independent. They probably aren't.
Plain-language summary
Addition rule (OR, for mutually exclusive events). If two events can't both happen, the probability that one or the other does is the sum.
Ecology example: A camera classifies a photo as bobcat, lynx, or cougar with probabilities 0.6, 0.1, 0.3. A photo gets exactly one label, so the classes are mutually exclusive. The chance it says "bobcat or lynx" is $0.6 + 0.1 = 0.7$ — and, equivalently, the chance it does not say cougar is $1 - 0.3 = 0.7$.
Plain-language summary
Bayes' rule (turning conditionals around). If you know P(positive test | disease) and the base rate of disease, you can compute P(disease | positive test).
Why this matters: for rare events, even a very accurate detector mostly produces false alarms. This is the math behind why "the eDNA was positive" doesn't necessarily mean "the fish is there."
Random variables in plain English
A random variable is a number that pops out of a random process. Examples:
- The count of fawns observed in a 1-km transect (an integer).
- Whether a tagged steelhead returns to spawn (1 or 0).
- The temperature of a stream measured next Tuesday (a positive real number).
For each random variable, two questions matter:
- What values can it take? Discrete (integers, categories) or continuous (real numbers)?
- How is its probability spread across those values? Concentrated near a mean? Heavy on zero? Skewed?
The answer to question 2 is the distribution. Different stories pick different distributions.
Biological intuition
Mental exercise. Before fitting any model, write the sentence "the random variable I'm modeling is..." and "the values it can take are..." and "the natural distribution given how the data were collected is...". If you can write all three confidently, you understand your model. If not, the modeling will go wrong somewhere.
The cast of characters (discrete distributions)
Bernoulli — one coin flip
Story: a single yes/no outcome. The only parameter is $p$, the probability of "yes."
Where you'll see it: Did the animal survive? Was the species present? Was the egg fertile?
Binomial — count of yeses in $n$ independent coin flips
Story: $n$ fixed-size Bernoulli trials with the same success probability $p$, all independent. You report only the total count.
Where you'll see it: Number of marten cameras that detected the species; number of nests that hatched out of $n$ monitored; number of deer testing positive for CWD out of a sample of 24.
Poisson — count of rare independent events in a window
Story: events happen "rarely" and "independently" at some average rate $\lambda$ per unit. The count in one unit is Poisson.
Where you'll see it: Whales per transect, calls per hour at an acoustic detector, mortalities per year, anything with an obvious "rate."
Key feature: mean equals variance. This will almost never match real data. Which leads us to...
Negative Binomial — counts that clump
Story: a Poisson, but the rate itself varies from unit to unit (heterogeneity). The marginal counts are over-dispersed.
Where you'll see it: Basically every count dataset in ecology. Use it as your default.
Multinomial — Binomial with more than two categories
Story: a fixed sample of size $n$ is sorted into $k$ mutually exclusive categories with probabilities $p_1, \dots, p_k$ that sum to 1.
Where you'll see it: Survey results (oppose/neutral/support); age-sex classifications; habitat-use proportions.
Key idea
Pattern to memorize: Each named distribution is identified by (a) its sample space — what values it can take — and (b) its parameters. Memorizing the story is far more useful than memorizing the PMF formula.
Common pitfalls (and how to avoid them)
Watch out
"My count data are Poisson because they're counts." Counts can be Poisson, but they very often aren't. Always check whether the variance is bigger than the mean. If it is, you need a Negative Binomial or a mixed model.
Watch out
Confusing P(A|B) with P(B|A). The probability of "positive test given disease" is not the same as "disease given positive test." Bayes' rule converts between them, and the answer depends on the base rate. (See the taxi puzzle.)
Watch out
Adding probabilities when they aren't mutually exclusive. $P(\text{rain Mon}) + P(\text{rain Tue})$ is
not $P(\text{rain Mon or Tue})$ unless one rules out the other. Use the inclusion-exclusion correction: $P(A \cup B) = P(A) + P(B) - P(A \cap B)$.
Watch out
Forgetting that independence is rare in ecology. Detections on the same transect, animals in the same herd, plots in the same site — all tend to be positively correlated. Treating them as independent inflates your effective sample size and gives you confidence intervals that are too narrow. This is the seed of the entire
mixed models conversation we'll have on Day 3.
Looking ahead
Tomorrow morning (Day 2) we shift from discrete to continuous distributions — Normal, Lognormal, Gamma, Beta, Exponential — and start to use them inside linear models. That same afternoon, we'll combine the linear model framework with the various distributions to get Generalized Linear Models, which are the bread and butter of modern ecological statistics.
Everything else this week — model selection, mixed models, MLE, Bayes — is built directly on the foundation we put down today.