**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) = ^{1}_{2} .

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, c_{0}, 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 = c_{0}.

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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