I need to cross check answers for the following question to do with probability

Question

I need to cross check answers for the following question to do with probability and distribution, if worked solutions could be provided so that I could check and/or compare where I have gone wrong that would be extremely helpful. The question is as following:

Sam scored an internship, helping a company write a bid to expand the shoulder of the Dalby highway (a 300km section of road, stretching from Dalby to st George). Sam is to calculate several probabilities below. For each task, Sam must:

i)identify the distribution,

ii)calculate the parameters of the distribution,

iii)calculate the probability by hand,

iv)and then confirm the probability in MATLAB.

Task 1: As part of expanding the highway, several mail boxes must be relocated. The local council has announced that one mail box can be expected every 25km. What is the probability that Sam observes 15 mail boxes while out counting them?

Task 2: Fortunately, not all mail boxes have to be relocated. There is a 20% chance that the mail box is already far enough back from the road. To get a maximum possible cost, what is the probability all 15 mail boxes have to be relocated?

Task 3: Data on the cost of relocating letter boxes on a previous job are listed in the data file attached. Given that the costs follow a normal distribution, calculate the probability that the cost/letter box on the current job will exceed $100/letterbox.

The data for Task 3 is the following:

At first, I just need to be able to check my answers against others to find out whether they are right, and if they arent, where I have gone wrong. If anyone is experienced with the use of MatLab, however, and could create working code to also cross check then that would be great. Thank you.

Explanation / Answer

Task 1:

(i) This distribution could be a Poisson distribution.

(ii) The parameter could be 1 mail box per 25km. So, for 300km the parameter will become 12 mailboxes.

(iii) If the the random variable X denotes the number of mailboxes observed.

The probability of observing 15 mailboxes while counting them is given by P(X=15) = e-121215/15! = 0.0724

(iv) x=15;

y=poisspdf(x,12);

Task 2:

(i) The distribution could be a Binomial distribution.

(ii) The probability of the mailbox beinng relocated is 20% i.e 0.20 and in total there are 15 mail boxes. So, the parameter of the distribution could be 0.20 and 15.

(iii) Let Y be a random variable which denotes the number of mail boxes that will have to be replaced. Then the probability that all 15 mail boxes will be relocated is P(Y=15)=15C15(0.20)15(0.80)0=3.2768 X10-11.

(iv) x=15;

y=binopdf(x,15,0.20);

Task 3:

(i) The given Probability distribution is a normal distribution.

(ii) The parameters of the distribution can be approximated using the given sample data.

The mean of the given data is = 94.41051

and the standard deviation of the given data is = 8.067

(iii) Let the random variable X denote the cost of replacing the mail box.The probability that the cost will exceed $100 is P(X>100)

Standardising the above probabilities we have P[Z > (100-94.41051)/8.067] =P(Z>0.69288) = 0.24419.

(iv) y=normpdf(100,94.41051,8.067)

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