All Matches

Solution Library

Expert Answer

Textbooks

Search Textbook questions, tutors and Books

Oops, something went wrong!

Change your search query and then try again

Result Not Found

Books FREE
Tutors
Study Help
Hire a Tutor
AI Study Help New

Search
Sign In
Register

study help
business
modern mathematical statistics with applications

Questions and Answers of Modern Mathematical Statistics With Applications

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
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
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
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).
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
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
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
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
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
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
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
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
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) +
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 ∩
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
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
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
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
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
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
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
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
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,
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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,
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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)
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
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
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
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,
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
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
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
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
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
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
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
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
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
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
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.
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
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
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
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
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.
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
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
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
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
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
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
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
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
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
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
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
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
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
Show that, for the ideal gas equation,∂P∂V∂V∂T∂T∂P= −1.
The ideal gas equation is PV = nRT. Calculate ∂P∂V and ∂P∂n.What are the scientific interpretations of these partial derivatives? Do they make scientific sense?

Showing 1400 - 1500 of 4596

First
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
Last

Services

Sitemap
Fun
Definitions
Become Tutor
Study Help Categories
Recent Questions
Expert Questions

Company Info

Security
Copyrights
Privacy Policy
Terms & Conditions
SolutionInn Fee
Scholarship

Get In Touch

About Us
Contact Us
Career
Jobs
FAQ
Campus Ambassador

Secure Payment

Download Our App

© 2024 SolutionInn. All Rights Reserved