random variables and probability distributions

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Random Variables and Probability Distributions. Modified from a presentation by Carlos J. Rosas-Anderson. Fundamentals of Probability. The probability P that an outcome occurs is: The sample space is the set of all possible outcomes of an event Example: Visit = {( Capture ), ( Escape )}. - PowerPoint PPT Presentation

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Random Variables and Probability DistributionsModified from a presentation by Carlos J. Rosas-Anderson1Fundamentals of ProbabilityThe probability P that an outcome occurs is:

The sample space is the set of all possible outcomes of an eventExample: Visit = {(Capture), (Escape)}

P(Capture) = # of captures/# of visits; P(Escape) = # of escapes/# of visits2Axioms of ProbabilityThe sum of all the probabilities of outcomes within a single sample space equals one:

The probability of a complex event equals the sum of the probabilities of the outcomes making up the event:

The probability of 2 independent events equals the product of their individual probabilities:

NOTE: Axiom 1 assumes that the outcomes in the sample space are mutually exclusive and exhaustive.Use example of randomly shuffled deck of playing cards with 52 cards.3Probability distributionsWe use probability distributions because they fit many types of data in the living world

Ex. Height (cm) of Hypericum cumulicola at Archbold Biological Station

4Probability distributionsMost people are familiar with the Normal Distribution, BUTmany variables relevant to biological and ecological studies are not normally distributed!For example, many variables are discrete (presence/absence, # of seeds or offspring, # of prey consumed, etc.)Because normal distributions apply only to continuous variables, we need other types of distributions to model discrete variables.5Random variableThe mathematical rule (or function) that assigns a given numerical value to each possible outcome of an experiment in the sample space of interest.

2 Types:Discrete random variablesContinuous random variablesThe Binomial DistributionBernoulli Random VariablesImagine a simple trial with only two possible outcomes:Success (S)Failure (F)

ExamplesToss of a coin (heads or tails)Sex of a newborn (male or female)Survival of an organism in a region (live or die)

Jacob Bernoulli (1654-1705)The Binomial DistributionOverviewSuppose that the probability of success is p

What is the probability of failure?q = 1 p

ExamplesToss of a coin (S = head): p = 0.5 q = 0.5Roll of a die (S = 1): p = 0.1667 q = 0.8333Fertility of a chicken egg (S = fertile): p = 0.8 q = 0.2

The Binomial DistributionOverviewImagine that a trial is repeated n timesExamples:A coin is tossed 5 timesA die is rolled 25 times50 chicken eggs are examinedASSUMPTIONS: p is constant from trial to trialthe trials are statistically independent of each otherThe Binomial DistributionOverviewWhat is the probability of obtaining X successes in n trials?

ExampleWhat is the probability of obtaining 2 heads from a coin that was tossed 5 times?

P(HHTTT) = (1/2)5 = 1/32The Binomial DistributionOverviewBut there are more possibilities:

HHTTTHTHTTHTTHTHTTTHTHHTTTHTHTTHTTHTTHHTTTHTHTTTHH

P(2 heads) = 10 1/32 = 10/32The Binomial DistributionOverviewIn general, if n trials result in a series of success and failures,

FFSFFFFSFSFSSFFFFFSF

Then the probability of X successes in that order is

P(X)= q q p q = pX qn XThe Binomial DistributionOverviewHowever, if order is not important, then

where is the number of ways to obtain X successes

in n trials, and n! = n (n 1) (n 2) 2 1n!X!(n X)! pX qn XP(X) =n!X!(n X)!The Binomial DistributionOverview

The Poisson DistributionOverviewWhen there are a large number of trials but a small probability of success, binomial calculations become impracticalExample: Number of deaths from horse kicks in the French Army in different yearsThe mean number of successes from n trials is = npExample: 64 deaths in 20 years out of thousands of soldiers

Simeon D. Poisson (1781-1840)The Poisson DistributionOverviewIf we substitute /n for p, and let n approach infinity, the binomial distribution becomes the Poisson distribution:

P(x) = e -xx!The Poisson DistributionOverviewThe Poisson distribution is applied when random events are expected to occur in a fixed area or a fixed interval of time

Deviation from a Poisson distribution may indicate some degree of non-randomness in the events under study

See Hurlbert (1990) for some caveats and suggestions for analyzing random spatial distributions using Poisson distributionsThe Poisson DistributionExample: Emission of -particlesRutherford, Geiger, and Bateman (1910) counted the number of -particles emitted by a film of polonium in 2608 successive intervals of one-eighth of a minuteWhat is n?What is p?

Do their data follow a Poisson distribution?The Poisson DistributionEmission of -particlesNo. -particlesObserved05712032383352545325408627371398459271010114120131141Over 140Total2608Calculation of :

= No. of particles per interval= 10097/2608= 3.87

Expected values:

2608 P(x) =e -3.87(3.87)xx!2608 The Poisson DistributionEmission of -particlesNo. -particlesObservedExpected05754120321023834073525525453250854083946273255713914084568927291010111144120113111411Over 1400Total26082608The Poisson DistributionEmission of -particles

Random events

Regular events

Clumped eventsThe Poisson Distribution

Review of Discrete Probability DistributionsIf X is a discrete random variable,

What does X ~ Bin(p, n) mean?

What does X ~ Poisson() mean?The Expected Value of a Discrete Random Variable

The Variance of a Discrete Random Variable

Continuous Random Variables If X is a continuous random variable, then X has an infinitely large sample spaceConsequently, the probability of any particular outcome within a continuous sample space is 0To calculate the probabilities associated with a continuous random variable, we focus on events that occur within particular subintervals of X, which we will denote as x

Continuous Random Variables

The probability density function (PDF):

To calculate E(X), we let x get infinitely small:Uniform Random Variables Defined for a closed interval (for example, [0,10], which contains all numbers between 0 and 10, including the two end points 0 and 10).

Subinterval [5,6]Subinterval [3,4]

The probability density function (PDF)Uniform Random Variables

For a uniform random variable X, where f(x) is defined on the interval [a,b] and where a

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