What is a probability distribution?
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What is a probability distribution?
It is the set of probabilities on a sample space or set of outcomes
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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.
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Practical Uses of Probability Distributions
• To calculate confidence intervals for parameters.
• Calculate critical regions for hypothesis tests.• How likely is a particular outcome?
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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
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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
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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
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Poisson
• Number of seeds falling in a gap• number of offspring produced in a season• number of prey caught per unit time
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Negative Binomial• Counts the number of failures before a
predetermined number of success occurs• Good at describing a patchy or clumped distribution
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Negative Binomial
• Variance can be larger than the mean (overdispersed)
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Geometric• The number of trials until you get a single
failure (or the number of failures until you get a single success)
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Beta-Binomial
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Continuous Distributions• Use probability density functions (pdf)
• Normal• Lognormal• Exponential• Gamma
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What’s the probability we had 2 inches of rainfall last night?
~20%
Probability density functions
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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?
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What is the probability that rainfall is between 1.98 and 2.25 inches?
= 1.98< X < 2.25
Probability density functions
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Area under the curve!
Probability density functions
= 1.98< X < 2.25
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Area under the curve!
100%
Probability density functions
=1
= 1.98< X < 2.25
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Normal DistributionAll real values
Add enough samples together and you get this bad boy=additive
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Normal Distribution Example:
Height of students
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Log-normalPositive real values
The product of many independent samples from same
distribution=multiplicative
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Population sizes in Deer
Log-normal example
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Gamma
The distribution of waiting times until a certain number of events take place
Positive real values
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Time till death of (α) crabs
Gamma example
α =1
What is the probability that there will be one crab death under 200 days?
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Exponential
The distribution of waiting times for a single event to happen
Positive real values
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Oyster survival
Exponential Example
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Distributions are often related to each other
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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
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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)
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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)
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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
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Conditional Probability
• Events are not independent from each other (dependent)
• The probability that event B will occur given that A has already occurred
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Example of conditional
• Infection status
Has lice
Does not have lice lice
Infected
Not Infected
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Example
• http://www.youtube.com/watch?v=mhlc7peGlGg
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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?
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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
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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?
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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?