Question A

Consider the following game of wheel of fortune:

The wheel can land on each segment and they are equally likely. Players win the amount of money labelled on the segment, with the modifiers (+€100 or double money) on the corresponding coloured segments.

  1. Write down the pmf and cdf of the outcome.
  2. Calculate the expected outcome.
  3. Calculate the variance of the outcome.
  4. What’s the most amount of money the casino can charge to spin the weel so that at least 30% of all players make a profit?
  5. What is the mean profit?
  6. If each contestant played the game \(n\) times and the prizes cumulated, what would you expect the distribution of the outcome to be if \(n\) was very very large?

Question B

Let X be the random variable representing the minimum value of two dice.

  1. Write down the pmf of X.
  2. Calculate \(\mathbb{E}[X]\) and \(Var(X)\).
  3. Find \(\mathbb{P}(X \leq 3)\).
  4. If \(Y\) is the is the minimum value of \(n\) dice thrown at the same time, what would you expect the distribution of \(Y\) to be if \(n\) was very very large?

Question C

It is known that the height of an appartment building in in Manhattan follows a Normal distribution, with mean 46.3m and standard deviation 10m. A building is chosen at random.

  1. What is the probability that it is between 36.3m and 56.3m?
  2. What is the probability that it is taller than 63m?
  3. What is the probability that it is shorter than 20m?
  4. What is the probability that it is between 40m and 55m?
  5. What is the probability that it is taller than 70m or shorter than 10m?
  6. If I choose a Manhattan resident at random, would I expect the mean height of their building to be 46.3m? Explain your answer.
  7. Give a reason why the height of Manhattan buildings cannot truly follow a Normal distribution.

Question D

It is known that the price of artworks sold at an auction in 2019 was Normally distirbuted with mean £1300 and standard deviation £250.

  1. Roughly what proportion of artworks were sold for more than £1300?
  2. Give the first and third quartiles of the distribution.
  3. What proportion of artworks were sold for more than £1000?
  4. What proportion of artworks were sold for more than £2000?