Probability distributions
A probability distribution describes the possible values of a random variable and how probability is assigned to them.
Discrete example
A binomial variable counts successes in \(n\) independent Bernoulli trials:
\[P(X=k)=\binom nk p^k(1-p)^{n-k}.\]Its mean and variance are \(np\) and \(np(1-p)\).
Continuous example
A normal variable is described by mean \(\mu\) and variance \(\sigma^2\):
\[X\sim N(\mu,\sigma^2).\]Biological choice
Counts, proportions, waiting times and continuous measurements have different mathematical properties, so the distribution should reflect how the data arise.
Key idea. A probability distribution is a mathematical model for uncertainty in a measured or generated quantity.