Discrete CDF table with constants

Given a discrete random variable with CDF values in a table containing unknown constants, use F(max)=1 and monotonicity to find the constants.

1 questions

Edexcel S1 2024 January Q7
  1. The cumulative distribution of a discrete random variable \(X\) is given by
\(x\)1234
\(\mathrm {~F} ( x )\)\(\frac { 1 } { 13 }\)\(\frac { 2 k - 1 } { 26 }\)\(\frac { 3 ( k + 1 ) } { 26 }\)\(\frac { k + 4 } { 8 }\)
where \(k\) is a positive constant.
  1. Show that \(k = 4\)
  2. Find the probability distribution of the discrete random variable \(X\)
  3. Using your answer to part (b), write down the mode of \(X\)
  4. Calculate \(\operatorname { Var } ( 13 X - 6 )\)