Find parameter n from mean

A question is this type if and only if it gives the mean (or expectation) of a discrete uniform distribution and asks to find the parameter n (number of values).

3 questions

AQA Further AS Paper 2 Statistics 2021 June Q3
2 marks
3 The random variable \(X\) has a discrete uniform distribution and takes values \(1,2,3 , \ldots , n\) The mean of \(X\) is 8 3
  1. Show that \(n = 15\)
    [0pt] [2 marks]
    LL
    3
  2. \(\quad\) Find \(\mathrm { P } ( X > 4 )\)
    3
  3. Find the variance of \(X\), giving your answer in exact form.
Edexcel S1 2003 November Q5
5. The random variable \(X\) has the discrete uniform distribution $$\mathrm { P } ( X = x ) = \frac { 1 } { n } , \quad x = 1,2 , \ldots , n$$ Given that \(\mathrm { E } ( X ) = 5\),
  1. show that \(n = 9\). Find
  2. \(\mathrm { P } ( X < 7 )\),
  3. \(\operatorname { Var } ( X )\).
Edexcel S1 Q7
7. The random variable \(X\), which can take any value from \(\{ 1,2 , \ldots , n \}\), is modelled by the discrete uniform distribution with mean 10 .
  1. Show that \(n = 19\) and find the variance of \(X\).
  2. Find \(\mathrm { P } ( 3 < X \leq 6 )\). The random variable \(Y\) is defined by \(Y = 3 ( X - 10 )\).
  3. State the mean and the variance of \(Y\). The model for the distribution of \(X\) is found to be unsatisfactory, and in a refined model the probability distribution of \(X\) is taken to be $$\mathrm { f } ( x ) = \left\{ \begin{array} { c l } k ( x + 1 ) & x = 1,2 , \ldots , 19
    0 & \text { otherwise } \end{array} \right.$$
  4. Show that \(k = \frac { 1 } { 209 }\).
  5. Find \(\mathrm { P } ( 3 < X \leq 6 )\) using this model.