Introduction to Stochastic Population Models

1 Introduction to Stochastic PopulationModelsThomas E. WehrlyDepartment of StatisticsTexas A&M UniversityJune 13, 20050-0 Mathematics 669 Contents1 Probability and Random The Basic Ideas of Probability .. Sample Spaces and Events .. Conditional Probability .. Independence.. Random Variables .. Probability Distributions of a Discrete .. Parameters of Probability Distributions .. Expected Values of Discrete RV .. Continuous Random Variables .. Percentiles .. Expected Values, Mean and Variance .. Joint Probability Distributions .. Conditional Distributions.. Conditional Distributions for Bivariate Continuous Some Special Cases .. Poisson Distribution .. The Poisson Process .. Normal Distribution .. Gamma Distribution .. Distribution of Elapsed Time in the Poisson Process .292 An Introduction to Stochastic Population Some Ecological Examples.

2 A Quick Contrast Between Deterministic and Stochastic Models Deterministic model .. Stochastic model ..32 Table of ContentsCopyrightc 2005 by Thomas E. WehrlySlide 0 Mathematics 6693 Basic Methods for Single Population Moments ofX(t).. Simulation of the Stochastic Process .. Kolmogorov Differential Equations for Probability Functions . Generating Functions.. PDEs for Cumulant Generating Functions ..504 Some Linear One- Population Linear Immigration-Death Models .. Deterministic model .. Stochastic model .. Application to the AHB Population Dynamics .. Linear Birth-Immigration-Death Models .. Solution to the Deterministic model .. Probability Distributions for the Stochastic model .. Generating Functions .. Application to AHB .. Simulation of the LBID Process ..675 Some Nonlinear One- Population Nonlinear Birth Death Models .. Deterministic model .

3 Probability Distributions for the Stochastic model .. Generating Functions and Cumulants .. Application to AHB Population Dynamics .. Nonlinear Birth-Immigration-Death Models .. Deterministic model .. Simulation of the Stochastic model .. Application to AHB Population Dynamics .. Summary of Single Population Models ..77 Table of ContentsCopyrightc 2005 by Thomas E. WehrlySlide 0 Mathematics 6696 Models for Multiple Compartmental Models .. The Deterministic Compartment model .. Stochastic Compartmental Models .. Basic Methods for Two- Population Models .. A Birth-Immigration-Death-Migration model .. Simulation of Predator-Prey model .. Simulation of a Competition model ..85 Chapter 1: Introduction to Probability and Random VariablesCopyrightc 2005 by Thomas E. WehrlySlide 0 Mathematics 6691 Probability and Random VariablesThe Models that you have seen thus far are deterministic Models .

4 Forany timet, there is a unique solutionX(t). On the other hand, Stochastic Models result in a distribution of possible valuesX(t)at atimet. To understand the properties of Stochastic Models , we need touse the language of probability and random The Basic Ideas of Sample Spaces and EventsProbability:Probability is used to make inferences about :Some process whose outcome is not known with Space:The collection of all possible outcomes of anexperiment or process; :Any collection of possible outcomes of an experiment; denotedA, B, Frequency Interpretation of ProbabilityA random experiment is carried out a large number (n) of times and thenumber (n(A)) of times that event A occurs is recorded. Then theproportion of times that A occurs will tend to the probability of A:n(A)n P(A)Chapter 1: Introduction to Probability and Random VariablesCopyrightc 2005 by Thomas E. WehrlySlide 1 Mathematics 669 Illustration of Long-Run Relative FrequencySuppose a die is tossed repeatedly, and we count the number of timesthat the toss results in six spots.

5 We then plot the proportion of timesthat the toss results in a six versus the number of (A)n(A) Frequency of Tosses of Die Resulting in a Sixnrelative 1: Introduction to Probability and Random VariablesCopyrightc 2005 by Thomas E. WehrlySlide 2 Mathematics 669 (A)>0for any (S) = 13. For any collectionA1, A2, ..of mutually exclusive events(Ai Aj= fori6=j),P(A1 A2 ) = i=1P(Ai)Properties: 0 P(A) 1 P( ) = 0 Probability an event does not occur:P(A ) = 1 P(A). P(A B) =P(A) +P(B) P(A B) IfAandBare mutually exclusive,P(A B) =P(A) +P(B)Chapter 1: Introduction to Probability and Random VariablesCopyrightc 2005 by Thomas E. WehrlySlide 3 Mathematics Conditional ProbabilityFor any two eventsAandBwithP(B)>0theconditional probabilityofAgiven thatBhas occurred:P(A|B) =P(A B)P(B)The multiplication ruleforP(A B)is:P(A B) =P(A|B)P(B)P(A B) =P(B|A)P(A)Law of Total ProbabilityLetA1, .. , Anbe mutually exclusive and exhaustive means thatA1 A2 An= also thatP(Aj)>0for eachj.

6 Then for any eventB,P(B) =n j=1P(B|Aj)P(Aj)IfP(B)>0, this law impliesBayes Theorem:P(Ak|B) =P(B|Ak)P(Ak) nj=1P(B|Aj)P(Aj)Chapter 1: Introduction to Probability and Random VariablesCopyrightc 2005 by Thomas E. WehrlySlide 4 Mathematics 669 Example:Diagnostic the two events:A= event that disease is presentB= event that diagnostic test is positiveWe usually know the following: Prevalence of disease, sayP(A) =.001 Sensitivity of test, sayP(B|A) = Specificity of test, sayP(B |A ) = want to know,P(A|B)orP(A |B )Solution:P(A|B) =P(B|A)P(A)P(B|A)P(A)+P(B|A )P(A )=( )( )( )( )+(1 )(1 )= (A |B ) =P(B |A )P(A )P(B |A )P(A )+P(B |A)P(A)=( )( )( )( )+(1 )( )= 1: Introduction to Probability and Random VariablesCopyrightc 2005 by Thomas E. WehrlySlide 5 Mathematics IndependenceTwo eventsAandBareindependentifP(A B) =P(A)P(B).They (A)>0andP(B)>0, this definition is equivalent toP(A|B) =P(A) andP(B|A) =P(B).Extension of Independence to Several EventsWe say thatA1, A2.

7 , Anaremutually independentif for everysubset{i1, .. , ik}(k 2), we haveP(Ai1 Ai2 Aik) =P(Ai1)P(Ai2) P(Aik)We say thatA1, A2, .. , Anarepairwise independentifP(Ai Aj) =P(Ai)P(Aj)for every pair(i, j), i6= independence does not imply mutual 1: Introduction to Probability and Random VariablesCopyrightc 2005 by Thomas E. WehrlySlide 6 Mathematics Random VariablesRandom variables help us to make a link between probability andnumbers that we observe as variable (rv)is a numerical valued function defined on asample space. A random variable X maps an outcome in a samplespace to a numerical probability that a rvXtakes a value in the setAis given byP[X A] =P[X 1(A)].We use capital letters such asXorYto denote random an elementary outcome. A value,X(s), ofXis random variable isdiscreteif it can take on a finite or countablenumber of variable takes on an uncountable number 1: Introduction to Probability and Random VariablesCopyrightc 2005 by Thomas E.

8 WehrlySlide 7 Mathematics Probability Distributions of a Discrete probability distribution of a discrete is a list of the distinct valuesxofXtogether with the associated probabilities:p(x) =P(X=x)ByP(X=x), we meanP(Ax)whereAx={s S:X(s) =x}.We can expressp(x)as a function or in a table:xx1x2x3..xkp(x)p(x1)p(x2)p(x3)..p( xk)A functionp(x)orpxis aprobability mass function (pmf)of somerandom variableXif p(x) 0allx allxip(xi) = 1 Chapter 1: Introduction to Probability and Random VariablesCopyrightc 2005 by Thomas E. WehrlySlide 8 Mathematics 669An alternative way to represent a probability distribution is by usingthecumulative distribution function (cdf):F(x) =P(X x) = y:y xp(y), < x < For a discrete random variable taking values onx1< x2< < xk,p(xj) =F(xj) F(xj 1), j= 2, .. , Parameters of Probability DistributionsSuppose that for each value of ,p(x; )is a probability distribution fora random variableX. Then is said to be aparameterof thedistribution.

9 The collection of distributions{p(x; ) : A}is called aparametric familyof 1: Introduction to Probability and Random VariablesCopyrightc 2005 by Thomas E. WehrlySlide 9 Mathematics Expected Values of Discrete RV Mean of a discrete RVThe mean of a rvXisE[X] = = x Dx p(x)whereDis the set of possible values ofX. Expected value of a function ofXThe expected value of a functionh(X)is:E[h(X)] = h(X)= x Dh(x) p(x)Ifh(X)is a linear function of the formaX+b:E(aX+b) =aE(X) +bChapter 1: Introduction to Probability and Random VariablesCopyrightc 2005 by Thomas E. WehrlySlide 10 Mathematics 669 Variance of a discrete variance of a discrete isV(X) = 2= 2X=E[(X )2] Thestandard deviationofXis = X= V(X) = 2= SD(X) The variance of a linear functionaX+bisV(aX+b) =a2V(X) =a2 2 Implications: V(aX) =a2V(X) SD(aX) =|a|SD(X) V(X+b) =V(X) SD(X+b) = SD(X)Chapter 1: Introduction to Probability and Random VariablesCopyrightc 2005 by Thomas E. WehrlySlide 11 Mathematics Continuous Random VariablesAcontinuous random variablecan assume any value in an interval onthe real line.

10 The distribution of a continuous random variable isdetermined by the probability density function (pdf). The pdf ofXis afunctionf(x)such that for any numbersaandbwherea < b,P(a X b) = baf(x)dxThe graph off(x)is often called a density of a PDFxpdfForf(x)to be a pdf it must (x) 0allx2. f(x)dx= 1(area under curve is 1).An alternative method of expressing the distribution of a continuousChapter 1: Introduction to Probability and Random VariablesCopyrightc 2005 by Thomas E. WehrlySlide 12 Mathematics 669random variable is using the cumulative distribution function (cdf). Thecdf of a continuous RV is defined as:F(x) =P(X x) = x f(y) of a of a CDFxcdfUseful Properties: P(a X b) =F(b) F(a) IfXis a continuous RV with pdff(x)and cdfF(x), then at everyxat whichF (x)exists:F (x) =f(x)Chapter 1: Introduction to Probability and Random VariablesCopyrightc 2005 by Thomas E. WehrlySlide 13 Mathematics PercentilesFor0 p 1the(100p)thpercentile of the distribution of acontinuous RVXis a valuexpsuch thatp=F(xp) Expected Values, Mean and VarianceThe expected value of a functionh(X)for a continuous rv is:E[h(X)] = h(x) f(x)dxSome special cases:Mean:E[X] = = x f(x)dxVariance:E[(X )2] = 2= (x )2 f(x)dxRemember:E[(X )2] =E[X2] (E[X])2= 2 Note:The properties of expectation and variance of linear functions alsohold in the continuous 1: Introduction to Probability and Random VariablesCopyrightc 2005 by Thomas E.