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modern mathematical statistics with applications

Mathematics And Statistics For Science 1st Edition James Sneyd, Rachel M. Fewster, Duncan McGillivray - Solutions

Let X be a random variable with the following probability function:Does X have a binomial distribution? Explain why or why not.
Let X ∼ Binomial(5, p) be the number of successes out of 5 independent trials. Denote a trial by S if it is a success, and by F if it is a failure. Consider the outcome SFFSS, in which the first trial is a success, the second trial is a failure, the third trial is a failure, and so on. Say
For each of the following scenarios, say whether or not it is reasonable to model X by a binomial distribution. If it is, specify any parameters you can, and any additional assumptions you might need to make. If it isn’t, say why not.a. X is the number of heads in 10 tosses of a coin.b. X is the
When a new disease like Covid-19 arises, doctors must experiment with different drug treatments on live patients. Suppose there are two drugs available, A and B. One of these is likely to be better than the other, but we don’t know which one it is. The doctors would like a strategy that ensures
An absent-minded professor owns one umbrella. Each morning he walks from home to work, and each evening he walks from work back home again. He only ever thinks to take the umbrella with him if it’s raining. Otherwise the umbrella stays where he last left it, whether that was at home or work. If
The Geometric distribution is a probability distribution that’s widely used in statistical modelling. We’ll see more details in Chapter 38. Here we look at just one example called Geometric(0.5). If you’re playing a series of games and you have a 50%chance of winning each game, the
A winning streak in sports describes a situation where a player wins several games in a row. Human intuition is remarkably poor when it comes to judging the chance of such winning streaks. We tend to underestimate how often a long run of successes will happen purely by chance. The famous Sports
Suppose X is a discrete random variable taking values 1, 2, . . ., 10, with probability function fX(), and cumulative distribution function FX(). Say whether each of the following statements is true or false.a. The values FX(x) must sum to 1 over x = 1, 2, . . . , 10.b. The values of FX(x) must all
Can you catch a liar? Do you think a machine can? Various liedetecting technologies are used by law enforcement agencies around the world. Here we look at a study (Hopkins et al, 2005)of voice stress analysis: a system that is claimed to distinguish liars from truth-tellers based on indicators of
The following table shows the hospitalisation rate for New Zealand Covid patients, broken down by age-group. For each age-group, the hospitalisation probability shown is the total number of hospitalisations divided by the total number of cases in that age-group.a. Is this a probability
In his famous pea-plant experiments, Gregor Mendel established principles of genetic inheritance that are now known as the Mendelian principles. One of these principles states that if you breed two parents who both have one dominant and one recessive allele for a genetic trait, the probability that
In humans, the proportion of right-handers is about 9 times the proportion of left-handers. Write down the probability distribution for handedness. (You may ignore the possibility of being ambidextrous.
Statistics on Covid-19 deaths gave the following breakdown by age-group for the first 1624 deaths in Italy in 2020:a. What is an appropriate sample space for studying the probability distribution represented by this table?b. What are examples of relevant events on your chosen sample space?c. Give a
What is the probability that a four-child family has exactly one boy? (Hint: you don’t need to write down the whole sample space to answer this question, but you should clearly state your definition of the sample space.)
Let Abe an event on a sample space Ω. For each of the following statements, say whether or not it is written in correct notation.a. A ∪ A = Ωb. P(A ∪ A) = 1c. P(A) ∪ P(A) = Ωd. P(A) + P(A) = 1e. P(A) ∪ P(A) = 1
Let A and B be events on a sample space Ω, and suppose P(A) = 0.6 and P(B) = 0.2.a. Find P(A) and P(B).b. Can you calculate P(A ∪ B), or do you need more information?c. Can you calculate P(A ∩ B), or do you need more information?d. What is the largest possible value of P(A ∪ B)?e. What is
A researcher wants to know if married couples are unusually likely to share the same star sign. They select married couples at random and record their star signs. There are 12 possible star signs.Say whether each of the following statements is true or false.a. A suitable sample space would be Ω =
Still using Ω = {people in your class}, let event F = {person is female} and T = {person is twenty years old}. Suppose I pick a 19-year-old female. For each of the following events, say whether or not it has occurred.a. T ∪ F.b. T ∪ F.c. T ∪ F.d. T ∩ F.e. T ∩ F.
Consider the random experiment "pick a random person in your class". The sample space is Ω = {people in your class}.For each of the following, say whether or not it is an event.a. H = {height of the person is less than 170cm}.b. T = {person’s height}.c. D = {person likes dogs}.d. P =
A whale-watch tour company wants to find the probability of seeing a whale on each of their trips. Which of the following is a suitable sample space for their investigation?a. Ω = {whale sightings}.b. Ω = {whale-watching trips}.c. Ω = {trips on which whales are seen}.d. Ω = {customers}.
An excitable system like we saw in Exercise 26.16 has a critical role to play in neurons, as it can filter out unwanted noise(i.e., random variability arising from random events at the level of individual ion channels). You want a neuron to respond to proper stimuli but you don’t want it to
The bistable equation is often used to model the front of a wave that is moving through space (such as, for example, a wave front of electrical activity moving across the surface of the heart, a moving front of rabies infection in a population of foxes, or the front of a forest fire).Systems like
We saw before how a simple equation for the cytoplasmic concentration of calcium,c, is dc dt= 0.01 −1.1c 20.32 + c 2.This is appropriate when the influx is constant and equal to 0.01 (with typical units of µM s−1). However, in some cases, for some cells, the influx is a periodic function,
In this question we’ll investigate some simple population models with periodic birth and death rates. Let’s begin with a typical population modelled with both birth and death. If N is the number in the population (measured in millions, say), then a typical model would be dN dt= rN − aN2,
If an object of mass m falls in a vacuum (in the earth’s gravitational field) it accelerates with acceleration g, where g = 9.88605 m s−2 is the gravity of Earth. Since acceleration Actually, gravity on the Earth’s surface varies by around 0.7%, from 9.7639 m s−2 on the Nevado Huascarán
Consider the initial value problem dy dt= (y − 1)2t − y +3 2with y(0) = 0.a. Find a value of y for which the solution to the differential equation is a constant. (This solution need not satisfy the initial value).b. Use Euler’s method with step size h = 1 to estimate the solution at t = 1,
Consider the initial value problem dy dt= (3 − y)(y + 1) with y(0) = 1.a. Use Euler’s method with step size h = 1 to estimate the solution at t = 1, t = 2 and t = 3.b. What will be the long-term behaviour of the solution as calculated by Euler’s method with step size h = 1?c. Is the long-term
Consider the following initial value problem dq dw= (q − 2) (w − q) with q(−2) = 1a. Find a value of q for which the solution to the differential equation is a constant. (This solution need not satisfy the initial value.)b. Use Euler’s method with step size h = 1 to estimate the solution to
In Exercise 25.5 you were asked to use qualitative methods to determine the long-term behaviour of solutions to the differential equation dy dt= e−√y sin(y).Check your answers to that exercise by solving the differential equation numerically.
If db dv+1 vb =sin(v)v 2, b(1) = 1, find b(2), b(3) and b(10).
Consider the same differential equation as in the previous question dy dt+ 2y = e t, y(0) = 1.a. Calculate the exact solution at t = 1, 2, 5 and t = 10.b. Now use Euler’s method, with h = 0.5, to calculate approximate values of y at t = 1, 2, 5 and t = 10.c. Calculate the error (i.e., the
Consider the differential equation dy dt+ 2y = e t, y(0) = 1.a. Calculate the exact solution y(2).b. Now use Euler’s method to calculate approximate values of y(2) using h = 1.0, 0.5, 0.25, 0.125 and h = 0.0625.Don’t do this by hand! Use a computer.c. Calculate the error (i.e., the difference
a. Use a numerical method (and a computer) to solve dR dt= sin(R), R(0) = 1, for 0 ≤ t ≤ 10, and plot the solution.b. What happens if you change the initial conditions? Can you see why?c. Can you predict the final value of R as t → ∞? (Hint:as t → ∞ the solution is getting flatter and
Suppose we have a reversible chemical reaction, where 2 molecules of A can react to form a single molecule of B, as shown in the chemical reaction diagram One common example would be the dimerisation of oxygen, where two oxygen molecules combine to form O2.A + A k1−→←−k2 B.If the reactions
Curtis (1986) proposed a model for the formation and repair of cellular DNA damage (called lesions) due to radiation. In Radiation damage to cells occurs, for example, in the treatment of cancer, or when a nuclear power station explodes and pours radiation into the atmosphere. Lethal lesions kill
A simple (but not entirely unrealistic) model of a non-lethal infectious disease assumes that the population is divided into two classes; S (those who are susceptible to being infected)and I (those who are already infected). When a susceptible meets an infected, the susceptible can become infected
When a strip of skeletal muscle is stimulated by an electric current it starts to develop tension. For example, if the muscle strip is being held at a constant length, it takes more and more force to hold the muscle at a constant length, as the muscle starts to pull. This force, which develops over
Another simple model for a neuron is the quadratic integrateand-fire model, which takes the form dV dt= V 2 + I(t), where V denotes the neuron’s membrane potential and I is some input, which we shall here assume to be constant. We’re going to use this example to show how a qualitative analysis
The theta model, often called the Ermentrout–Kopell model, is one of the simplest models of neuronal bursting oscillations.The model is Bard Ermentrout and Nancy Kopell are two of the most eminent theoretical neuroscientists alive today.The variable θ represents the neuron’s membrane
So far we have learned a number of techniques for solving a differential equation that is given to us. But very often, in the real world, we need to infer the underlying differential equation by looking at the data. This can be a much more difficult task.We illustrate this by using Covid-19 data
Another differential equation that is often used to model population growth is the Gompertz equation, dn dt= αn ln K n, for some positive constants α and K. Some typical experimental data, together with a two fits to the data (using the Gompertz model and the logistic model), are shown in Fig.
Consider again the logistic differential equation from the previous question, dn dt= kn 1 −n K.Without solving the equation (i.e., using a qualitative analysis)In other words, if the population can grow, it will, until the maximal population is reached. However, if the populations starts out
The logistic differential equation dn dt= kn 1 −n K, n(0) = n0, is one of the most common equations used to model population This is the logistic equation, which is discussed in more detail in Section 4.1.1. growth and death.a. Show that the solution of the logistic differential equation is
A model as simple as the ones you’ve seen in this chapter was used to tell us something important about the HIV virus that causes AIDS. One intriguing thing about HIV infection is that patients infected with the virus can go many years with no symptoms. Since the amount of HIV in their bodies
Radiocarbon dating is a method of dating organic material (animal bones, petrified trees, etc) by measuring the amount of radioactive carbon (14C) they contain. All living things continually incorporate carbon (C) into their bodies, a certain fraction of which is 14C, the rest being mostly 12C.
Consider the differential equation dy dt= −y 2.a. Find a one-parameter family of solutions (i.e., a formula for solutions, with one arbitrary constant in the formula).b. Check that you have the correct answer in (a) by substituting your answer back into the differential equation.c. Calculate the
For each of the following differential equations, draw the phase line, find all the equilibria, and state their type (sink, source, or node).a.dy dt= (1 − y)(y + 4).b.dy dt= (1 − 2y)(4 − y 2).c.dy dt = (6 − y 2 )y 2 .
Consider the differential equation dp dy= p(p 2 − 1) − k, where k is a constant that can be varied.a. What are the equilibria when k = 0 and are they sinks, sources, or nodes?b. At what values of k are there exactly two equilibria? For each value of k for which there are exactly two equilibria,
Consider a differential equation du dt= p(u), where p(u) has the graph shown in Fig. 25.13.a. At what values of u does this differential equation have equilibrium points?b. Sketch the phase line for this differential equation.c. For each interval bounded by equilibrium points, describe the
Consider the differential equation dy dt= 2t y 2 + y 2.a. Find a one-parameter family of solutions to the DE (i.e., a formula for solutions, with one arbitrary constant in the formula).b. Check that you have the correct answer in (a) by substituting your answer back into the DE.c. Find a solution
Use separation of variables to solve ydy dt= y 2sin t − sin t.
a. Find the steady states of the differential equation df du= f 3 − µ f , where µ is a constant, and determine their stability.b. Sketch all the steady states as functions of µ, drawing sinks as solid lines and sources as dotted lines.c. Find all the places where the number of steady states
a. Find the steady states of the differential equation dx dy= x 2 + k,where k is a constant, and determine their stability.b. Sketch all the steady states as functions of k, drawing sinks as solid lines and sources as dotted lines.Diagrams like this, where the steady states are plotted as functions
Consider the differential equation dy dt= (1 + y)(1 − y).a. What are the steady states?b. Using a computer package such as Wolfram Alpha or Matlab, can you find an exact solution (i.e., can you find the solution in the form y(t))?c. Using any method you like, determine what happens to the
As of the time of writing this book, Covid-19 is one of the most serious infectious diseases currently in existence, and has caused a global pandemic, killed hundreds of thousands of people (most likely millions) and caused enormous harm to the world’s health and economy. Data from Australia are
Suppose you measured the speed of a falling shuttlecock and got the following data points.a. Use a Riemann sum to approximate the total distance travelled by the shuttlecock.Use whichever Riemann sum you want.b. Use the trapezoid rule to approximate the total distance travelled by the
The flow of a river can be measured by the discharge, Q, which is the amount of water passing through a cross-section per second. Q can be calculated as Q =∫ B 0v(b)h(b)db, where h(b) is the depth (in m) at point b across the river, v(b)the flow velocity (in m/s) at that point, and B is the
A flow injection analysis (FIA) experiment relies on analyte flowing through a series of tubes before reaching the detector.An analyte is a substance whose chemical constituents are being identified and measured.The tubes are useful for introducing reagents to cause colour changes, mixing, or
Chromatography allows for the separation and quantification of complex mixtures, and depends on the fact that different compounds adhere to a liquid with different strengths. For example, if you(a) fill a tube with a stationary beads coated with some liquid,(b) put some complex mixture in at one
The flux, j, through an ion transporter was measured as a function of time, t, and the following data were obtained.a. Plot the data and use the trapezoid method to estimate the total amount of water that flows down the river during these 25 days.Make sure your total amount of water is given in the
The flow, F, in a river was measured as a function of time, t, and the following data were obtained.a. Plot the data and use the trapezoid method to estimate the total amount of water that flows down the river during these 25 days.Make sure your total amount of water is given in the correct
Electrical current comes out of our wall plugs (well, in New Zealand anyway) as alternating current (AC), which means that the current and the voltage vary like a sine wave, with a root mean square voltage of 230 V. But what does the root mean square (RMS) voltage mean? In this exercise we take a
At the beginning of this chapter, on page 439, we saw the equation for measuring cardiac output by the dye-dilution method.If an amount A of a dye is injected into the heart, and its concentration, c(t), measured in the aorta, then the cardiac output, F, is given by F =A∫ T 0c(t) dt.T is some
In Exercise 17.7 we saw that the photocurrent of a salamander rod can be approximated by the function R(t) = R0t ne−kt, where t is in units of s, and R is in units of pA. For simplicity, just set R0 = k = 1 and let n = 3.You definitely want to compute these integrals on a computer, not by hand.a.
Here’s an interesting conundrum. When you place a heavy brick on a table and leave it there, the table does no work. Sure, it has to exert a force to keep the brick up, but this force isn’t moving through a distance, so no work is done. The table will happily hold the brick there for an
How much work is done in gradually compressing 0.5 moles of an ideal gas at 400 K from 2 L to 1.5 L? Would you get a different answer if you expressed the volume in units of m3?The answer has to be no, of course, but why not?
Take one mole of an ideal gas at standard temperature and pressure. Now quickly decrease the pressure to 90 kPa and hold it constant.a. What is the new volume of the gas?b. How much work has been done during the expansion of the gas?c. Instead of decreasing the pressure quickly and holding it
Show that you can calculate work by multiplying the average force by the distance travelled.
The force, F, needed to stretch or compress a spring a distance x (away from its rest position, which is defined to be x = 0) is given by F = k x for some constant k, called the spring constant. The sign of F This is called Hooke’s law. It’s reasonably accurate for a lot of springs(and even for
A constant force of 200 newtons was used to move a brick, and 150 joules of energy was expended. How far was the brick moved?
a. James lives close to the sea in Auckland, New Zealand.Suppose he lifts a book (of mass 0.5 kg) from the floor of his office and puts it on a bookshelf 1 m high, how much work has he done?For this question you will have to look up the acceleration due to gravity, at sea level in Auckland, on the
In Fig. 5.2 we saw how the concentration of Ca2+ions in an airway smooth muscle cell oscillates, and this oscillation is shown in detail in Fig. 5.17. From this second figure we calculated that the calcium concentration,c, can be approximately described by the function Suppose that c has units of
Find the area underneath the straight line p = mt +c, for any constants m andc, over the interval [a, b]. How else could you work this out? Make sure your two methods give the same answer.b. What is the average velocity of the object over the interval [1, A], for any constant A > 1.c. What is the
Find the area underneath the curve J = sin(θ) over the interval[0, 2π].
What is the area underneath the curve of k(b) =b, from b = 2 to b = 3? Work this out two different ways, and show that you get the same answer each time.
a. Calculate ∫ 11 v2 dv. Take the limit as → 0, and thus calculate ∫ 1 01 b2 db.b. Calculate ∫ 11 vdv. Take the limit as → 0, and thus calculate ∫ 1 01 bdb.c. Calculate ∫ 11√v dv. Take the limit as → 0, and thus calculate ∫ 1 01√b db.d. Calculate ∫ 1 1 v k dv, for
a. Calculate ∫ n 11 v2 dv. Take the limit as n → ∞, and thus calculate ∫ ∞1 1b 2 db.b. Calculate ∫ n 11 tdt. Take the limit as n → ∞, and thus calculate ∫ ∞1 1u du.c. Calculate ∫ n 11 tk dt, where k > 1. Take the limit as n → ∞, and thus calculate ∫ ∞1 1u k du.
Calculate the average values of each of the following functions between the two given points. Where you can, give a geometrical interpretation of the average value.a. g(n) = 2 between -2 and 2.b. h(s) = s between -2 and 2.c. v( f ) = f 2 between 0 and 1.d. Q(θ) = sin(θ) between −π and π.e.
The flux, j, of Na+ions through a Na+channel into a cell was measured as a function of time, t, and the following (pretend)data were obtained.a. Plot the data.It’s always a good idea to look at your data first, just to get a general idea of its shape and behaviour.By the way, setting n(0) = 0 is
a. Use Riemann sums to find an upper bound and a lower bound for ∫ 0−2 t3 dt. How would you refine these estimates(i.e., make the upper bound closer to the lower bound so that you get a better estimate for the actual area)?b. How would you use Riemann sums to find upper and lower bounds for ∫
a. Why is a right Riemann sum for an increasing function always an overestimate of the area?b. Why is a right Riemann sum for a decreasing function always an underestimate of the area?
Compute the left and right Riemann sums for ∫ 2 0√2 − w dw using four intervals. Average the left and right Riemann sums and compare to the actual area (which is 4√2/3). Why do you get a different result than the previous question?
Compute the left and right Riemann sums for ∫ 1 0(2−z) dz, using four intervals, and compare them to the exact value (which you can calculate from a geometrical argument). Now average the left and right Riemann sums. How does the average compare to the actual value?
Consider the function k(u) = |u|.a. Calculate ∫ 0−2 k(u) du exactly. Hint: don’t use any integration methods to do this, just use some geometry and the formula for the area of a triangle.b. Estimate ∫ 0−2 k(u) du by using a right Riemann sum with four rectangles. Show that this right
In Section 21.3 we approximated the area under the curve of y = x 2, from x = 0 to x = 5 by using the function values on the right-hand sides of the approximating rectangles. Calculate two more approximations (as drawn in the figures in Section 21.3), one using the function values on the left-hand
One of the most important applications of this theory is in the area of curve fitting (often called linear or nonlinear regression).This topic is covered in detail in Part XII of this book; here we consider only a simple example to give a flavour of how the method works, and how it involves finding
When a pre-synaptic neuron fires at time t = 0, then the postsynaptic neuron receives a stimulus that is often described by the function If you’re not sure what pre-synaptic and post-synaptic neurons are, have a look at Wikipedia. s(t) = Ate−t.However, the constant A sometimes isn’t a
The Gaussian function in two dimensions is G(x, y) = Ae−(x−µ)2σ2 x−(y−ν)2σ2 y , where A, σx, σy, µ and ν are constants. Find any stationary points of G(x, y). Plot G and thereby classify any stationary points you find.
A box with no top has to have a volume of 9 m3. However, the material used for the base of the box costs twice as much as the material used for the sides of the box. What dimensions must the box have in order to minimise the total cost?
Suppose you wanted to build a box with a volume of 9 m3, but with no top. What are the dimensions of the box that use the least amount of cardboard?
A box with six sides has to have a volume of 1 m3. What shape should the box be in order to minimise the surface area of the box?These first three exercises are simple examples of optimisation, a branch of mathematics and statistics that is used in almost every area of science, and many areas of
Find the stationary points of the function U(i, p) = sin(i 2 + p 2).Draw the surface of U and from this surface identify which of the stationary points are maxima and which are minima.
Find the stationary points of the function M(a,b) = 2a 3 +6ab2−3b 3−150a, and identify the extrema by drawing a graph of the surface. (Hint: this is a bit tricky. You will probably need to fiddle with the vertical scale of the graph to make all the humps appear clearly. Then you may need to
above (c(x, t) is a solution to the diffusion equation). Suppose that you want to calculate the value of c at x = 1±0.002 and t = 1±0.005. What are the absolute and relative uncertainties in the calculation of c? How does the relative uncertainty of c compare to those of t and x?
Consider again the function c(x, t) =1√4πt e−x 24t , the same one that we used in Exercise
Absolute and relative uncertainties of functions can behave in nonintuitive ways. Let y = e x.a. Calculate the absolute uncertainty of y in terms of the absolute uncertainty of x. What happens as x gets large and negative? Explain why this makes sense.Hint: plot the graph of y(x). What does this
Suppose that the function W(x, t) is defined by W(x, t) = f (x + ct), where f (s) is any function (of one variable s, where s = x +ct)f can’t actually be any function at all, it has to be reasonably nice, but we won’t worry about that detail for now. and c is a constant.a. Show that This
Newton’s law of gravitation says that the force, F, between two objects of mass m1 and m2, a distance d apart, is given by F =Gm1m2 d2, where G = 6.674×10−11 N m2/kg2 is the gravitational constant.a. Calculate ∂F∂m1,∂F∂m2 and ∂F∂d. Which of these are positive and which are
At standard temperature and pressure (i.e., 273.15 K and 100 kPa) one mole of an ideal gas has a volume of 22.711 L. For the rest of this question, assume that all the calculations are done for one mole of an ideal gas at standard temperature and pressure.a. If the pressure is increasing at a rate
Suppose that, for an ideal gas, P, V and T are all functions of time, t. By taking partial derivatives of the ideal gas equation show that P0(t)P+V 0(t)V=T 0(t)T.Now show the same result by first taking the log of the ideal gas equation, and then differentiating.

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