Modelling assumptions and refinements

A question is this type if and only if it asks about the assumptions needed for a discrete uniform model, comments on their validity, or suggests refinements to the model.

3 questions

Edexcel S1 2005 January Q6
6. A discrete random variable is such that each of its values is assumed to be equally likely.
  1. Write down the name of the distribution that could be used to model this random variable.
  2. Give an example of such a distribution.
  3. Comment on the assumption that each value is equally likely.
  4. Suggest how you might refine the model in part (a).
Edexcel S1 Q1
  1. Thirty cards, marked with the even numbers from 2 to 60 inclusive, are shuffled and one card is withdrawn at random and then replaced. The random variable \(X\) takes the value of the number on the card each time the experiment is repeated.
    1. What must be assumed about the cards if the distribution of \(X\) is modelled by a discrete uniform distribution?
    2. Making this modelling assumption, find the expectation and the variance of \(X\).
    3. (a) Explain briefly why, for data grouped in unequal classes, the class with the highest frequency may not be the modal class.
    In a histogram drawn to represent the annual incomes (in thousands of pounds) of 1000 families, the modal class was \(15 - 20\) (i.e. \(\mathrm { f } x\), where \(15000 \leq x < 20000\) ), with frequency 300 . The highest frequency in a class was 400 , for the class \(30 - 40\), and the bar representing this class was 8 cm high. The total area under the histogram was \(50 \mathrm {~cm} ^ { 2 }\).
  2. Find the height and the width of the bar representing the modal class.
AQA Further AS Paper 2 Statistics 2020 June Q3
2 marks
3 The random variable \(X\) represents the value on the upper face of an eight-sided dice after it has been rolled. The faces are numbered 1 to 8 The random variable \(X\) is modelled by a discrete uniform distribution with \(n = 8\)
3
  1. Find \(\mathrm { E } ( X )\)
    3
  2. \(\quad\) Find \(\operatorname { Var } ( X )\)
    3
  3. Find \(\mathrm { P } ( X \geq 6 )\)
    3
  4. The dice was rolled 800 times and the results below were obtained.
    \(\boldsymbol { x }\)\(\mathbf { 1 }\)\(\mathbf { 2 }\)\(\mathbf { 3 }\)\(\mathbf { 4 }\)\(\mathbf { 5 }\)\(\mathbf { 6 }\)\(\mathbf { 7 }\)\(\mathbf { 8 }\)
    Frequency1036384110744185240
    State, with a reason, how you would refine the model for the random variable \(X\).
    [0pt] [2 marks]