Competency 2: Describe the concepts, processes, and tools required to conduct comprehensive health assessments for individuals, families, communities, and populations. Describe the data necessary to make an informed community health assessment. Explain how data helps determine the health care needs of a community. Describe a strategy for obtaining data needed for a community health assessment. o Explain how to establish the validity and reliability of data used in a community health assessment. • Competency 3: Explain the internal and external factors that can affect the health of individuals, families, communities, and populations. o Explain the factors that affect the health and wellness of a community. o Explain how to obtain information on the factors that affect the health and wellness of a community. .Prove the Cauchy distribution is subexponential and strong subexponential. 3.2.UsingdirectestimatesforconvolutionoftheParetodensity,showthePareto distribution is subexponential and strong subexponential. 3.3. Using direct estimates for convolutions, prove that any regularly varying at infinity distribution is subexponential and strong subexponential. 3.4.LetF andGbetwodistributionsthatareregularlyvaryingatinfinity,andlet 0 < p < 1. (i) Prove that the distribution pF +(1−p)G is regularly varying at infinity. (ii) Prove that the distribution F∗G is regularly varying at infinity. 3.5. Prove by direct calculations of convolution density that the exponential distribution is not subexponential. 3.6.Supposethatξ ,...,ξ areindependentrandomvariableswithcommondis1 n tributionF.Provethemaximum,max(ξ ,...,ξ ),hassubexponentialdistributionif 1 n and only if F is subexponential. +3.7. Suppose that F and G are weakly tail-equivalent distributions on R . Prove thattheconvolutionsF∗F andG∗Gareweaklytail-equivalenttoo.Provethesame for n-fold convolutions, for any n≥3. 3.8. Suppose that ξ , ..., ξ are independent random variables with a common 1 n α1exponential distribution. Find the asymptotics for probability P{ξ +...+ξ > αn n1 x}as x→∞if (i) allα > 1; i (ii) allα < 0; i (iii) allα 6∈[0,1]. i 3.9. Let non-negative random variable ξ have the Weibull distribution with the β− xtail F(x)=e ,β >0.Forwhichvaluesofα >0,doesP{ξ+ξ >x}∼P{ξ >x} α as x→∞? 1−α α αHint: Make use of the equalityξ +ξ =ξ (1+ξ ) and Problem 2.23. 3.10.Letrandomvariableξ havethestandardlog-normaldistribution.Forwhich values of parameterα > 0, doesP{ξ +ξ > x}∼P{ξ > x}as x→∞? α 3.11.How do Pitman’s criteria work (i) for the Pareto distribution; (ii) for the exponential distribution? 3.12.Specify Kesten’s bound for the standard Cauchy distribution. 3.13.LetX(t)beacompoundPoissonprocesswithasubexponentialdistribution ofjump.Foreveryt,findtheasymptotictailbehaviourofthedistributionofX(t) in terms of the jump distribution. 3.14. Find an example of subexponential distribution F and of counting random variableτ such that the equivalence ∗τF (x)∼EτF(x) as x→∞ doesn’t hold. Hint: Make a link to Galton-Watson process.