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

Prob

abili

ty

# 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 DistributionAll real values

Add enough samples together and you get this bad boy=additive

Normal Distribution Example:

Height of students

Log-normalPositive 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

• http://www.youtube.com/watch?v=mhlc7peGlGg

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 looklike?