MA10212代写、代做Lab 3 CJ、代做R程序语言、R代写代写

日期：2019-03-22 09:06

MA10212: Computer Lab 3 CJ, Mar 2019

For homework: Problem 1

Freddy Fisher is planning to spend 5 hours on the river bank catching as

many fish as he can. Let T1 denote the waiting time (in minutes) until he

catches his first fish, T2 the time between catching fish 1 and fish 2, and

so forth. We shall assume the Ti

, i = 1, 2, . . ., are independent and each

Ti ～ Exp(0.1), where 0.1 is the rate parameter.

(a) (4 marks) Write code to simulate values T1, . . . , T100, the first 100

between-fish intervals. The intention is that there are more than enough Tis

to describe what happens in 5 hours of fishing. Check this assumption holds

by finding the sum T1 + . . . + T100 and noting that it is a lot higher than 300

(you should get an answer in the range 800 to 1200, or thereabouts).

Now find the times X1, X2, . . . , at which fishes are caught (in minutes since

Freddy started fishing), setting X1 = T1, X2 = T1 + T2, etc. You may

wish to start by defining a vector y of length 100 and using the command

y[n]=sum(t[1:n]) to put the sum of the first n elements of the vector

t = (T1, . . . , T100) in element n of y. After completing this process, you

should find that some values y[n] exceed 300 — and you need to remove

these. Experiment with the commands y < k and x = y[y<k] for a specified

constant k to discover a neat way of selecting the elements of y that you want

to retain. Check that your final vector x contains the correct information.

(b) (2 marks) Create a function sample.x that simulates one sequence

of the times at which fish are caught. The command x=sample.x() should

produce a vector x containing the sequence of fish-catching times. Call your

function to check that it produces sensible looking results.

(c) (5 marks) Write code to generate 1000 realisations of Freddy’s 5 hour

fishing trip. In each case note the total number of fish caught in the five hour

period and call these numbers N1, . . . , N1000.

Madame Cholet insists that the number of fish caught should follow a

Poisson distribution. Let μ be the mean of the observed values N1, . . . , N1000.

Compare the histogram of your sample of values N1, . . . , N1000 with that of

a sample from a Poisson distribution with mean μ to see if the data agree

with Madame Cholet’s theory.

Continue this investigation in an appropriate manner so that you can give

a firm conclusion as to whether or not Madame Cholet’s claim is correct.

Present appropriate graphical displays to support your argument. (You may

find it useful to refer back to the methods used in Lab Sheet 2, Problem 2.)

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MA10212: Computer Lab 3 CJ, Mar 2019

(d) (3 marks) Generate a new sample of 1000 realisations of Freddy’s

fish-catching adventure. In each case note W, the time Freddy had to wait

between the first and second fish that he caught. Put the values of these

waiting times W in the vector Wsample.

What distribution do the data in Wsample appear to follow? Does this agree

with your expectations?

(e) (6 marks) Now generate a new sample of 1000 realisations of Freddy’s

fish-catching expedition. This time find V , the length of the between-fish

interval that contains the mid-point of Freddy’s 5 hour stay at the river bank.

So, if fish are caught at times X1, . . . , XN , you need to find V = Xi+1 ? Xi

where i is such that Xi < 150 and Xi+1 > 150. (You might experiment with

the commands max(x[x<k]) and min(x[x>k]) for a specified constant k to

find a neat way to carry out this calculation.) Put the values of these interval

lengths V in the vector Vsample.

What distribution did you expect the data in Vsample to follow? Is this the

case? Can you find a distribution that fits the data in Vsample?

Compare the means of the data in the vectors Wsample and Vsample. Think

about how these interval lengths arise and give a plausible explanation for

why the data in Vsample should be so different from the values in Wsample.

(f) (2 marks) You are asked to advise on the design of a study into the

provision of healthcare services. The investigators plan to analyse data from

a National Health Service data-base to see how long people have to wait

between the time their doctor decides they need a hip replacement and the

time the hip replacement operation is performed. The proposal is to extract

information about their waiting times for all patients who were in the process

of waiting for a hip replacement on 1 July 2017.

Given what you have observed in the fish-catching example, what comments

would you make to these investigators?

Presentation (3 marks)

Your submission should be well organized and easy to follow.

Tidy up your output by deleting any errors and unnecessary commands.

Give each plot a title and label the x and y axes.

Answer the questions as stated and use colour to highlight your answers and

distinguish these from output produced directly by R.