What is a probability distribution? It is the set of probabilities on a sample space or set of outcomes

A random variable is a variable (typically represented by x) that has a single numerical value that is determined by chance. A probability distribution is a graph, table, or formula that gives the probability for each value of the random variable.

Practical Uses of Probability Distributions To calculate confidence intervals for parameters. Calculate critical regions for hypothesis tests. How likely is a particular outcome?

Definitions Discrete Distributions – The outcomes are a set of integers – Describe counting or sampling processes – Ranges that include some or all of the nonnegative integers Continuous Distributions – A probability distribution over continuous range of values

Binomial Each trial can only have one of two values – black/white, yes/no, alive/dead Probability # of successes p =0.1 p =0.5 p =0.8

Poisson Gives the distribution of the number of individuals, arrivals, events, counts, etc. in a given unit of counting effort Use Poisson when the number counts could be limitless

Poisson Number of seeds falling in a gap number of offspring produced in a season number of prey caught per unit time

Negative Binomial Counts the number of failures before a predetermined number of success occurs Good at describing a patchy or clumped distribution

Negative Binomial Variance can be larger than the mean (overdispersed)

Geometric The number of trials until you get a single failure (or the number of failures until you get a single success)

Beta-Binomial

Continuous Distributions Use probability density functions (pdf) Normal Lognormal Exponential Gamma

What’s the probability we had 2 inches of rainfall last night? ~20% Probability density functions

Precision for continuous variables ! 0% chance of getting exactly 2 inches 0% Probability density functions What’s the probability we had 2 inches of rainfall last night?

What is the probability that rainfall is between 1.98 and 2.25 inches? = 1.98< X < 2.25 Probability density functions

Area under the curve! Probability density functions = 1.98< X < 2.25

Area under the curve! 100% Probability density functions =1 = 1.98< X < 2.25

Normal Distribution All real values Add enough samples together and you get this bad boy =additive

Normal Distribution Example: Height of students

Log-normal Positive real values The product of many independent samples from same distribution =multiplicative

Population sizes in Deer Log-normal example

Gamma The distribution of waiting times until a certain number of events take place Positive real values

Time till death of (α) crabs Gamma example α =1 What is the probability that there will be one crab death under 200 days?

Exponential The distribution of waiting times for a single event to happen Positive real values

Oyster survival Exponential Example

Distributions are often related to each other

Probability and Rules To understand ecological models need to understand basic probability Define: All possible outcomes that could occur Frequency that certain outcomes occur Probability of an even happening = # of ways it can happen/Total # of outcomes Sum of all the probabilities is always 1

Mutually Exclusive events If event A happened then event B cannot happen at the same time A or B = Prob(A U B) Prob(A or B) = Prob(A) + Prob(B)

Joint Probability Want to know the probability that two events will occur together at the same time Probability bear will catch male fish larger than 30cm P (A or B) = P(A) + P(B) – P(A&B)

Independent events Event A has no influence on event B Multiply the probabilities to find the combined probability of a series of independent events Not independent

Conditional Probability Events are not independent from each other (dependent) The probability that event B will occur given that A has already occurred

Example of conditional Infection status Has lice Does not have lice lice Infected Not Infected

Example lGg lGg

Example The number of red mites counted on each of 150 apple leaves Suppose that each mite had an equal probability of finding itself on a leaf, irrespective of the number of other mites present on a leaf How would a random distribution of mites over the leaves appear?

Example Burrow survey before and after cattle grazing Recorded if a burrow entrance was open or collapsed Compared the pre-grazing condition to the post grazing condition

Example 300 male minks Interested in growth rates between 5 different colors of mink Brown color type is thought to have a very rapid growth rate What distribution would like fit the body weight of brown minks vs. time?

Example Sage-grouse populations are estimated by attendance at leks You survey males and females at all the leks in Idaho for one breeding season What would you expect the distribution to look like?