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Let X have the pdf where the distribution of ? is Find the

Probability and Statistical Inference | 9th Edition | ISBN: 9780321923271 | Authors: Robert V. Hogg, Elliot Tanis, Dale Zimmerman ISBN: 9780321923271 41

Solution for problem 4E Chapter 6.9

Probability and Statistical Inference | 9th Edition

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Probability and Statistical Inference | 9th Edition | ISBN: 9780321923271 | Authors: Robert V. Hogg, Elliot Tanis, Dale Zimmerman

Probability and Statistical Inference | 9th Edition

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Problem 4E

Problem 4E

Let X have the pdf

where the distribution of θ is  Find the compound distribution of X, which is called the Burr distribution.

Step-by-Step Solution:
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Margaret S Biology 1060 (General Biology II) Week 14 Lecture Notes Unit 4: Ecology 4-­‐11-­‐16 What is Ecology • The relationship between organisms and their environment (between individual organisms, and between multiple organisms and the environment) o Ex. A bird building a nest in a tree: relationship between bird and tree is ecology o Ex. Our bodies are ecosystems for other organisms: the interaction between microbes and us is ecology • Limits to populations: Population ecology o Thomas Malthus, 1790: 1 census of U.S., population had doubled in 25 years which wouldn’t work in Britain (Britain’s population couldn’t conceivably double in 25 years); this only works in the U.S. because at that time there was a lot of uncolonized space (once we murdered all the Native Americans because we were horrible) o This study influenced Darwin: lead him to believe resources are limited for all species, which is when natural selection kicks in (the most adapted individuals live, the least adapted individuals die) o Populations are the fundamental units of evolution § Populations evolve; not individuals Population Ecology • Population ecology is the study of how and why a population changes over time • Populations are described by 3 key factors: o Size o Geographic range o Spatial distribution (density): it can be equal within the range or clustered in certain areas • To measure the size of a population: o Census: count everyone; organisms must be easy to observe, not too numerous, and live in an easily defined area o Sampling: sessile (stationary) organisms; count how many are in a square foot, then multiply by the number of square feet in the area the population lives (often done with plants) o Mobile organisms: mark-­‐and-­‐recapture (collect a certain number of organisms, label them, and recapture the same number later on to see how many are new and how many are recaptured) • The specific math behind these methods is not important for the test Population Ecology Behind Krill • The estimated population is 800 trillion • They live off the coast of Antarctica • The population density diagram shows an unequal density, and the average is 42 million krill per kilometer squared • These statistics will likely change because that is the nature of population statistics o Ex. An increase in the number of predators will cause population size to decrease, then the predator population size will decrease after the krill decrease in number o Ex. Krill need cold water to live in, but if water warms (climate change) the population size may decrease Spatial Distribution • Random: no pattern to dispersal of individuals • Clustered: clumped areas of organisms with empty spaces in between o Ex. Could be clustered because of clustered resources • Dispersed (regular): equal space between individual organisms o This happens with territorial animals o This also happens with plants that release chemicals that prevent growth of anything else nearby • Could start with 1 distribution type, then change over time o Ex. Shrubs may grow together in their youth in clumps for survival; start being more random when competition starts; and end up equally spaced by maturity Population Dynamics • Birth rate (positive contributor to population size) vs. Death rate (mortality rate: negative contributor to population size) • Immigration rate (positive; individuals coming into population) vs. Emigration rate (negative; individuals leaving population) • We can predict population growth over time using mathematical models o Imagine bacteria on a petri dish (no immigration/emigration) o T1-­‐T0 = ΔT (T=time, N= population size) o Graph: (T0, N) (T1, N1) (T2, N2) o All time intervals are equal o Change in population size from T0 to T1: N1-­‐N = ΔN o Growth rate from T0 to T1: N1 – N = ΔN T1 – T0 ΔT • Per capita growth rate T0 to T1 (takes into account starting population size): o (ΔN/ΔT)/N, where N = initial population size • Modeling population growth: o T0 = 10 o 1 hour later (T1) = 20 o Per capita growth rate (r) o (ΔN/ΔT)/N = ((20-­‐10)/1)/10 = 1 • Modeling bacterial growth o N = 10, r = 1 o Nt = N1(1+r)^T o What will the population size be at… § T + 1 10 x 2 = 20 § T + 2 20 x 2 = 40 § T + 3 40 x 2 = 80 § T + 4 80 x 2 = 160 § T + 5 160 x 2 = 320 § T + 6 320 x 2 = 640 o Exponential growth curve: the per capita growth rate (r) is constant o Population Growth • Population growth: ΔN = (b-­‐d) + (i – e) o b = births o d = deaths o i = immigration o e = emigration • Per capita growth rate: r = (ΔN/ΔT)/N • When unlimited growth is allowed, the growth curve will be exponential • Exponential growth is independent of population size • The population with the highest growth rate has the curve that increases the fastest • Exponential growth of conifer (Scots pine): pollen accumulation rate in lake sediments from different time points are sampled to measure population growth • Assumptions of exponential growth: o r is constant over time, no immigration or emigration, resources are unlimited o Because of these assumptions, exponential growth is not realistic; populations will eventually reach carrying capacity (K) because resources are not ever unlimited 4-­‐13-­‐15 Factors that Cause Mortality Differ Throughout the Life Cycle • Survivorship curves plot the proportion of the cohort alive at the end of each stage 3 Major Types of Survivorship Curves • Type 1: (humans): survivorship curve is really high, most individuals approach the maximum life span • Type 2: (songbirds): constant survivorship, an individual is not particularly more likely to die earlier/later in life • Type 3: (plants): death rate is high at the start of life, survivorship is high later in life Life History Strategies • BirthàGrowthàReproduceàDeath • Tradeoff between reproduction and lifespan, and number of offspring and investment per offspring o r-­‐selected species maximize their reproductive role; they produce many offspring and don’t invest much in them o K-­‐selected species live at their carrying capacity of environment; they produce fewer offspring but invest in them more 4-­‐15-­‐16 r-­‐Strategists and K-­‐Strategists • r-­‐Strategists have lots of offspring, with low survival (ex. Mosquitos and most insects) • K-­‐Strategists have few offspring with high survival (ex. Elephants) • K-­‐selected strategists stay around carrying capacity (they thrive at K, a value of population size that is sustainable in their environment) • r-­‐selected strategists fluctuate a lot more, they just maximize r • Oak trees are K-­‐selected because their acorns are large and survive longer • Giant clams are r-­‐selected Metapopulation • A metapopulation is a group of populations linked by migration • They are dynamic; they undergo a lot of extinction and recolonization but the extinction is local (one group becomes extinct, so its easy for another group to recolonize the same area) • They can be naturally occurring or artificial (we create metapopulations when we destroy large habitats) • When the species are isolated into patches, they will have more species going extinct; but when “corridors” connect the patches of habitat, less species go extinct than in the isolated patches, though there will still not be as many populations as in the original large group • Wildlife overpasses on roads provide corridors in forests, etc. Review Questions will be posted this weekend (4-­‐22)

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Chapter 6.9, Problem 4E is Solved
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Textbook: Probability and Statistical Inference
Edition: 9
Author: Robert V. Hogg, Elliot Tanis, Dale Zimmerman
ISBN: 9780321923271

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