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modern mathematical statistics with applications

Questions and Answers of Modern Mathematical Statistics With Applications

For each of the following functions show that the second-order mixed partial derivatives are equal (i.e., the order of differentiation doesn’t matter). If you need something extra to do, check the
If one of your fellow students tells you that they discovered a function, p(b, g), such that pb = 2b + 4g and pg = 3b + 5g, do you believe them? If that’s the correct expression for pb, what would
If u(x) = x 2, v(t) = e tand G(u, v) = sin(u(v)2 + v 2), calculate dG dt .
If u(t) = t 2, v(t) = t + e tand G(u, v) = sin(u 2 + v 2), calculate dG dt .
If u(y) = sin(4y), v(y) = e y + y 2and G(u, v) = e ue v, calculate dG dy .
When an advantageous gene appears in a population, or when an infectious person appears in a population of susceptibles, a traveling wave can often appear. In the first case, the advantageous gene
Consider the contour map of Rangitoto Island in Fig. 18.9.How can you tell where the peak of the island is? How can you tell which are the steep parts? Can you sketch the basic shape of the island
In the previous question you plotted a Gaussian function G(x, y). In this question we look at contour plots of G.a. Set µ = ν = 0 and σx = σy = 1. Show that the contour plot of G is a series of
The function By the way, µ is the Greek letter, mu, not the Latin letter u, and ν is the Greek letter nu, not the Latin letter v.It’s easy to confuse them.G(x, y) = e−(x−µ)2 2σ2 x
Suppose you take a bright red dye, and release a drop (at time t = 0) in a long, thin, horizontal tube of water. You then watch the drop spread to the right and to the left. You intuitively expect
We saw in Section 4.1 how the logistic equation is used to describe a variety of things. Another example of logistic growth is shown in Fig. 17.14, which plots the density of a population of E. coli
The Lennard–Jones potential describes the intermolecular potential energy, V, as a function of the distance,r, between a pair of particles. It’s probably the most widely used simple model of
When a population (such as a fish species) is harvested, the maximal sustainable yield, or MSY, is the harvesting rate that gives the greatest yield, without driving the population to extinction. If
Ice floats on water because it has a lower density than water.If d and T are, respectively, the water density and temperature, This is a really unusual property.Other substances whose density
Optimal foraging theory is a theory that animals spend the optimal amount of time collecting food from a single food patch. The theory is not without controversy, but it does seem to be a valuable
The rate, V, of an enzyme reaction is often described by the Michaelis–Menten equation, V =VmaxS K + S, where Vmax and S are constants. This is an equation we’ve used a lot in this book. However,
On page 305 we briefly considered a solution to the diffusion equation, which models how a pollutant, for example, diffuses from a single concentrated release. If you measure the concenIn other
It is shown in the theory of gravitational attraction that a wire bent in the form of a circle of radius a exerts a force, f , upon a particle in the axis of the circle (i.e., in the line through the
An undergraduate student is tossed into the air by a rampaging bull in Pamplona. The distance of the student from the ground is described by the function s(t) = 1 + 14t − 5t 2, where t is time in
Nuclear magnetic resonance (NMR) is a common method for determining the chemical composition of samples. Different compounds absorb electromagnetic radiation at a frequency characteristic of the
A Gaussian function, G(x), has the form G(x) = e−(x−µ)2 2σ2.Find the first derivative of this function and hence show that there is a stationary point at x = µ.
If you present a light flash to a salamander rod then the electrical current generated by the rod first rises and then decays, as shown in Fig. 17.12. The red curves are a very simple model.If R is
Hankinson’s equation describes the compressive strength of wood, as a function of the angle, relative to the grain of the wood, at which that stress is applied.Suppose that σ0 is the compressive
Is it possible to have a function that has no maximum or minimum?
Where are the critical points of the quadratic z(u) = au2 +bu+c? What kind of critical points are they?
Find a function, M(u), that has M00(0) = 0 but where u = 0 is not an inflection point.
In Exercise 4.22 we saw the equation, due to Fisher et al (1943), S = α ln 1 +Nα, where S is the number of species in a random sample from a population, N is the number of individuals in the
If you connect two resistors in parallel, the total resistance, Rt, over the circuit is given by 1Rt=1 R1+1 R2.If R1 is decreasing by 2 Ω/s but R2 is increasing by 3 Ω/s, what is the rate of
Suppose that an ideal gas (for which PV = nRT, remember)is changing its pressure and volume at a constant temperature and number of moles. That is, P and V are changing while n and T are held
When hydrogen ions (H+) enter a cell they quickly bind to various proteins, called buffers. This is so that the cell can control its pH precisely.Interestingly, the same is true of Ca2+ions, whose
The head length at age x, H(x) (in m), of the blue whale is related to the total length at age x, T(x), (also in m) by the equation This is an example of an allometric equation, which are discussed
Consider the curve defined implicitly by xy 2 + y − 2 = 0.Calculate dy dx when x = 1. Check your answers by plotting the curve.
What is the relationship between the rate of growth of the area, A, of a circle, and the rate of growth of the radius, r, of that circle? When r = 1 what is A 0(t)r 0(t)?
The Folium of Descartes is described by the equation x3 + y 3 − 3axy = 0, for some constanta. Just for convenience, set a = 1. Find all This is one of the famous equations in mathematics, and
Find all the points on the graph s 2 + 2st − 3t 2 = 1 where the tangent line is horizontal. Check your answer by plotting the graph.
Find all the points on the graph h 3 − 27h = b 2 − 90 where the tangent line is vertical. Check your answer by plotting the graph.
A circle of radius 1 has the equation p2 + e 2 = 1.What is the slope of the circle when p =1 2? Sketch the graph and its tangents at p =1 2.
Saccharomyces cerevisiae is a species of yeast that has been used in baking and brewing for thousands of years, and so it has been the subject of much scientific research over the last 100 years or
In Fig. 12.1 at the beginning of Chapter 12 we saw a graph showing how the rate of a calcium ATPase pump varies as a function of calcium concentration. If we extract some of the data we get the
A strong base was titrated against 10 mL of strong acid and the following table contains part of the data read by a pH probe.a. Plot the data.b. Numerically calculate the first derivative of pH at
The rate of growth of a population was measured every year for 10 years, and the following data were obtained.a. Use the data points at y = 3 and y = 5 to estimate R 0 (y) at y = 4.b. Similarly,
In 2002, Krahn et al published a report for the US Department of Commerce, studying the number of Southern Resident Killer Whales in the US Northwest (Krahn et al, 2002). Their data, shown in Fig.
Consider again the distance/time plots of the two sprinters shown in Fig. 13.5 at the beginning of Section 13.2. If we let x denote distance (in m), and t denote time (in s) then each sprinter’s
The Weber–Fechner law says that the relationship between Gustav Fechner was a student of Ernst Weber and named the law after his mentor, in recognition of the experimental work done by Weber that
The normal distribution, or Gaussian, is G(x) =1√2πσ2 e−(x−µ)2 2σ2, where σ > 0 and µ are constants. Where is G(x) increasing?Where is G(x) decreasing?
The Hill equation for the rate, V, of a cooperative enzyme reaction is V(S) =VmaxS nKn + S n, where n > 1 is usually an integer.a. What is V(0)? What is V(Kn)?b. Show that V 0(S) is sometimes
The Michaelis–Menten equation for the rate, V, of an enzyme reaction is V(S) =VmaxS K + S Show that V 0 (S) is always decreasing and that V 0 (S) → 0 as S → ∞.
Consider again the equation for queue length we first saw at the beginning of Section 12.4. Just to remind you, if λ is the average rate of arrivals at the queue, and µ is the average serving time,
Suppose that when you throw a stone upwards its distance, x(t), measured in metres from the ground after t seconds is given bya. What is the stone’s average velocity between t = 0 and t = 2
The Gompertz equation is N(t) = N0e Aα(1−e−αt), for some (positive) constants N0, A and αa. Calculate N 0 (t) and N 00(t).b. As you will learn in Chapter 17, the maximum of N 0 is at a place
The logistic equation is often used to model population growth(see Section 4.1.1); the number, N, in the population grows as a function of time according to N(t) =N0 1 + e−kt .a. Choose nice simple
A simple model for a quantum particle is the time-independent one-dimensional Schrödinger equation~2 2m d 2ψ(x)dx2 = Eψ(x).Here x represents distance, and ~ is equal to h 2π where h is Planck’s
When you take a single dose of an antibiotic, or other kind of medication, the concentration, C (in µg/mL), of drug in your blood stream is often well described by the equation C(t) = C0te−kt, for
What is the nth derivative of ln(x)?
What is the nth derivative of sin(2x)?
What is the nth derivative of e 2x?
In Fig. 13.5 we showed the data for how many seconds it took two sprinters to run 10 m segments of a 100 m sprint. The data for Florence Griffith-Joyner are given in Table 13.12.a. From these data
When you take a single dose of a drug (such as aspirin or paracetamol), the concentration of the drug in your blood decays along an exponential curve, at least approximately. Thus, if C(t) is the
Suppose that z(t) =√4 t. Using the limit definition of the derivative, show that z 0(t) does not exist at t = 0.
Suppose thatU(i) =√i. Using the limit definition of the derivative, calculate U 0(i).
Suppose that C(y) = cos(y). Using the limit definition of the derivative, calculate C 0(y).
Suppose that A(s) = s +1 s. Using the limit definition of the derivative, calculate A 0(1).
Suppose that M(v) =−5 v. Using the limit definition of the derivative, calculate M0(4).
Suppose that J(p) = 3p 2. Using the limit definition of the derivative, calculate J 0(4).
In this question we will show that the derivative of g(x) =cos(ax) is g 0(x) = −a sin(ax), using the limit definition of the derivative, i.e., using g0(x) = lim h→0 g(x + h) − g(x)h.a. First,
Follow these steps to calculate the value of the derivative of H(r) = e rat the point r = 1.a. Calculate the gradient H(r+h)−H(r)h(with r = 1) for different values of h, and complete the following
In Exercise 4.22 we saw the equation, due to Fisher et al (1943), S = α ln 1 +Nα, where S is the number of species in a random sample from a population, N is the number of individuals in the
Imagine using your muscle power to lift a mass. If the mass is small, you can lift it quickly, as the force exerted on the mass by gravity is small. If the mass is larger, you would only be able to
When you take a single dose of a drug such as paracetamol, the concentration of the drug in your blood is described by some function of time, P(t), say. Without specifying P(t) exactly, sketch P(t)
A hydrogen molecule comprises two hydrogen atoms held together by their bond; if the molecule receives enough energy to overcome that bond the molecule will dissociate. The amount of energy needed is
The behaviour of an electron in a hydrogen atom can be described by the radial function R(r) = Nr a02 e−r a0 , where N > 0 is a constant and a0 > 0 is the Bohr radius(another constant). For
As we saw in Exercise 4.23 the Gompertz equation, N(t) = N0e Aα(1−e−αt),can be used to model growing populations. Here, N is the number in the population, and A, N0 and α are positive
Population density, N(t), as a function of time, t, can be described by the logistic equation (see Section 4.1.1)N(t) =K 1 +K N(0)− 1e−r t.The initial population is N(0), and K and r are
Photoreceptors are cells in the retina that change their voltage in response to light. These changes in voltage are relayed to the brain via the optic nerve, and thus we see things. In response to a
Calcium ATPase pumps are proteins that sit in the membrane of a cell and use energy in the form of molecules of adenosine triphosphate (ATP) to remove Ca2+from the cell, against a steep concentration
Suppose that a cell membrane receptor is stimulated (by a hormone, say, or a neurotransmitter) at time t = 0, and that the concentration of activated receptor, R∗, is given by the following
The rate,r, of many enzyme reactions is related to the substrate concentration, S, by the Hill equation r =V Sn Kn + S n, for some constants n (a positive integer), V and K. What is the horizontal
What is lim f →∞ke−Af 2 eB f ?You will have to consider four different cases, as A and B can each be either positive or negative.
Calculate f (x) = x 10e−x for the following values of x.a. x = 1.b. x = 10.c. x = 100.d. x = 1000.Using your results, infer the value of limx→∞ x 10e−x. Can you guess the value of limx→∞
Calculate f (x) = xe−x for the following values of x.a. x = 1.b. x = 10.c. x = 100.Using your results, infer the value of limx→∞ xe−x.
Calculate f (x) = e−x for the following values of x.a. x = 1.b. x = 10.c. x = 100.Using your results, infer the value of limx→∞ e−x.
What happens to the function sin(1 x) as x → 0? (Watch out.The answer to this is a bit tricky.)
Can you explain to yourself and to others what the following would mean.a. lim x→∞f (x) = ∞.b. lim x→∞f (x) = −∞.c. lim x→−∞f (x) = −∞.
Can a function have 3 horizontal asymptotes? Can a function have 3 vertical asymptotes?
For each of the following scenarios give an example function f (x) which satisfies the conditions provided, and sketch f (x)around this limiting behaviour.a. lim x→−∞f (x) → ∞, where f (x)
Consider the reaction of ferric nitrate with ammonium carbonate, i.e.,α Fe(NO3)3 + β (NH4)2CO3 −→ γ Fe2(CO3)3 + δ NH4NO3.a. Write an equation that relates α and γ such that there are the
Methane burns in oxygen to form carbon dioxide and water.The reaction is CH4 + x1O2 −→ x2CO2 + x3H2O for some undetermined coefficients x1, x2, and x3.a. By balancing the number of H, C or O
The reaction of dichromate (Cr2O 2−7) with hydrogen ions (H+)and ferrous ions (Fe2+) is given by the reaction x1Cr2O 2−7 + x2Fe2+ + x3H+ −→ x4Cr3+ + x5Fe3+ + x6H2O.We have to determine the
Suppose a student performed an experiment to determine the concentration of a substrate as a function of time (with units of s). Denote the substrate concentration by S, with units of mol.The student
Suppose we have a biochemical reactiion where the concentration, s, of substrate is known to be s(t) = ke−At2 eBt, for some constants k, A, and B. Here, t is time (in units of seconds) and s has
The absorbance of a molecule, A, is proportional to its moNote that µg/mL is equivalent to ppm; this is a common unit in atomic absorption spectroscopy(AAS) which is usually done at low
For our final problem using Leslie matrices we use some real data on a wild rabbit population in England (Smith and Trout, 1994). Their motivation for the study was to determine how best rabbit
Suppose that a particular endangered species of fish has three distinct stages of life, each lasting one year. Each fish begins life as a hatchling (H), then becomes a juvenile (J), then an adult
Suppose that we have a population that, in year n, has xn population of juvenile females and yn population of adult females, but this time we assume that the juvenile females can also reproduce, just
Suppose that we have a population that, in year n, has xn population of juveniles and yn population of adult females.The units of xn and yn could be, for example, in millions of individuals (which
The PageRank method is a lot more complex than simply ranking all the pages according to the number of incoming links.Illustrate this using the network in Fig. 11.7. What ranking would you get if you
Suppose you have a very small internet of 4 web pages, each of which links exactly once to each other web page. Which web page has the highest page rank? What about if you have a slightly larger
Solve the page ranking problem in the previous chapter (Exercise 10.8) to rank the web pages in order of popularity.
Muhtar et al (2021) published a Leontief input-output model of integrated farming in Karehkel village in Indonesia. Their model of integrated farming had four compartments; vegetables(mostly lettuce,
On page 227 we saw an example of a Leontief input-output problem in economics, based on 1947 data from the USA. The answer we got in that example wasn’t all that interesting, but it becomes more

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