3.1 Statistical Distributions

Random Variable Observation = Variable Outcome = Random Variable Examples: – Weight/Size of animals – Animal surveys: detection outcome seen/not seen?

Probability Distributions Used to describe patterns of variability in outcomes (e.g., # of times detected, weight) Depends on the attributes of a random variables (RV) Ex: Weight: – Continuous – Necessarily positive – Most in narrow range with some large and small values Use mathematical function to describe variation in a set of values

Attributes of R.V. => Type of Distribution Discrete or continuous – Discrete: survival, reproductive status, litter size, population size, sex, observed/not observed etc. – Continuous: weight, length, rainfall, temperature, etc. Range of possible values? (limited range?) Mean – average value Variance – spread in values Skewness, kurtosis – shape of the variability (skewed, Peak vs. Tails)

probability mass (density) function Discrete – the probability of each discrete value occurring (pmf) Continuous – probability of the value occurring within an interval (pdf) Must sum to one

Specification of a distribution

Statistical distributions most commonly used in our field

Bernoulli Distribution

let p = 0.8 f(x=0|p=0.8) = 1-p = 0.2 and f(x=1|p=0.8) = p = 0.8 E(x) = 0.2* *1 = 0.8 Variance = 0.2*(0.8^2) + 0.8*(0.2^2) = 0.16 Example

Binomial (Multiple Bernoulli trials) Consider groups of Bernoulli r.v.’s (N = 3) Possible outcomes 000,100,010,001,011,101,110,111 0/3 : (1-p) (1 way) 1/3 : p(1-p) 2 100, 010, 001 (3 ways) 2/3 : p 2 (1-p) 011, 101, 110 (3 ways) 3/3 : p (1 way) Combinations: number of ways of “choosing” the outcomes Total # of successes is a binomial random variable

Binomial Distribution – Parameters: bin(n, p) n independent trials p = probability of success at each trial – Possible outcomes 0,1,2,….n X= number of successes – The same as n independent Bernoulli(p) – Combination term occurs in the function (X are 1, n-X are 0) – E(X) = np – Var(X)= p(1 – p)n

Example n = 10 nesting sea turtles captured in year 1 X = number that are captured the following year S = probability caught in the next year (survival?)

Binomial distribution n!n! x! (n – x)! S x (1–S) (n–x) Pr(X = x) =

Example A better way to think about the distribution of the number of turtles recaptured (x) will be where – S = survival – F = fidelity to population, given survival –  breeding probability, given faithful – p = probability captured, given breeder.

Multinomial Distribution Number of times each of > 2 discrete possible outcomes occur in N trials Example: rolling a die 6 possible outcomes ; π i = 1/6 Parameters: N and π i ’s Function looks similar to the binomial – combinatorial term and probability

Examples, with N = 3 trials: Probability of getting three 1’s – – 3!/3!0!0!0!0!0! = 1 Result: P(3,0,0,0,0,0) = 1 x (1/216) = 1/216 Probability of one 1, one 4,and one 6 – 3!/1!0!0!1!0!1! = 6 Result: P(1,0,0,1,0,1) = 6 x (1/216) = 6/216

Poisson Distribution Often used for counts (e.g., number of eggs in a nest, # of animals in a plot) Possible values are all non-negative integers λ is the only parameter E(x) = λ and V(x) = λ

Normal Distribution Extremely common and useful distribution, with location (μ) and shape (σ²) parameters Why is this useful? – Many variables are normally distributed – Central limit theorem: For large n’s, means tend to be normally distributed, even if not originally normal – Estimates of parameters intuitive E(x) = μ Var(x) = σ 2

Standard normal pdf: Any normal can be converted to a standard normal = Standardization of R.V.: (X – μ)/σ Normal Distribution

Other useful distributions Discrete: – Negative binomial (less restrictive than Poisson) Continuous – Χ 2 and F distributions – used for test statistics – Uniform – all values in an interval have same probability – Beta – flexible distribution for values between 0 and 1 – Gamma – flexible distribution for values ≥ 0

To remember Observations = Random Variables Distribution => Describes Variability Parameters  describes distribution

Uses of Probability Distributions Frequency of possible outcomes: Frequency of values (observed) for samples from a population (e.g., weights, # survived and dead) Probability of specific outcomes: e.g., Pr. of observing a marked animal during 3 consecutive samples Uncertainty in prediction: e.g., the expected number of animals that will survive over the next winter is between 70% and 80% Uncertainty in knowledge: Description of our “belief” about the value of a parameter (e.g., the expected proportion of males at birth is almost certainly between 0.45 and 0.55)