Instructions to students:

All questions are compulsory.

Materials:

Dictionaries are not permitted

Graph Paper

TASK: AS1101 Probability & Statistics 1 [CT3a] Assignment-City, University of London

QUESTION 1

An exam paper provides students with a data set consisting of 24 times (in hours and minutes) and instructs them to illustrate it by means of a suitable diagram. Figures 1a and 1b below are two of the diagrams submitted.

a)Criticize the two diagrams.[4 marks]

b)Assuming the numerical values in figure 1b are correct (with the stem in hours and leaves representing minutes), draw a cumulative frequency diagram.[3 marks]

c)Calculate the median and quartiles of the data set and show how they can be derived from your diagram[3 marks]  [Total: 10 marks]

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QUESTION 2

A is the event that I have at least one lecture during a day, B the event that I have at least one committee meeting, C the event that I have at least one meeting with a student and D the event that the first entry in my diary is for a lecture. If I have a committee meeting, it is always the first event in the diary for that day, and I never arrange meetings with students on days when I have committee meetings.

(i) Draw a Venn diagram to illustrate the situation.     [2 marks]

(ii) Explain why B ∩ D ⊂ C ∩ D ⊂ A ∩ D.       [2 marks]

a)In a five-day week there are an average of 3 days when I have lectures, 3 days when I have meetings with students and 1.5 days when I have committee meetings.   Suppose that A and B are independent, A and C are independent, and P(C|D) = ¹₂ .

b)In a 10-week term, on how many days do I expect to have no appointments at all?

c)Calculate the largest possible value for the probability that my day begins with a lecture and involves at least one meeting with a student.    [6 marks]    [Total: 10 marks]

QUESTION 3 : AS1101 Probability & Statistics 1 [CT3a] Assignment

1.I want to take a train to Edinburgh but all seats are booked. I am told that empty seats occur according to a Poisson process with rate λ = 0.5 per hour.

• Let T denote the time, in hours, until the first empty seat is available. Write down the density function of T and use it to calculate P(2 < T ≤ 3).
• Use the Poisson distribution to calculate the probability that there are no empty seats in the first two hours but at least one in the third hour.  [5 marks]

2. A function F is defined asi)Show that there is one value, c₀, of c for which F is the cumulative distribution function of a continuous random variable. Evaluate the density function f corresponding to F when c = c₀.

ii)Suppose X is a random variable with density function f . Calculate the expectation of (1−X)³.  [5 marks] [Total: 10 marks]

QUESTION 4

Andrew and Bob play the following game.

• Both players flip three fair coins and their score is given by the resulting number of heads.

• After seeing his score, but unaware of Bob’s score, Andrew can decide to swap his score with that of Bob.

Andrew adopts the the following strategy: he will retain his score only if it is 2 or 3 and swap it if it is 0 or 1.Denote by A and B Andrew’s and Bob’s final scores

i)Compute the probability of the following events: {A = 3 ∩ B = 3}, {A = 2 ∩ B = 1}, {A = 0 ∩ B = 2}.[5 marks]

ii)Compute the conditional expected value of Andrew’s score given the number of heads in his coin flip. Deduce Andrew’s expected score [3 marks]

iii)Andrew changes his strategy and decide to keep his score only if he sees 3 heads in his coin flip. Will his expected score increase? [2 marks]  [Total: 10 marks]

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