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.

2 questions · Moderate -0.3

5.02b Expectation and variance: discrete random variables
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Edexcel S1 2024 January Q7
10 marks Moderate -0.3
  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 \(\operatorname { Var } ( 13 X - 6 )\)
Edexcel S1 2024 October Q6
Moderate -0.3
  1. A biased die with six faces is rolled. The discrete random variable \(X\) represents the score which is uppermost. The cumulative distribution function of \(X\) is shown in the table below.
\(x\)123456
\(\mathrm {~F} ( x )\)0.10.2\(3 k\)\(5 k\)\(7 k\)\(10 k\)
  1. Find the value of the constant \(k\)
  2. Find the probability distribution of \(X\) A biased die with eight faces is rolled. The discrete random variable \(Y\) represents the score which is uppermost. The probability distribution of \(Y\) is shown in the table below, where \(a\) and \(b\) are constants.
    \(y\)12345678
    \(\mathrm { P } ( Y = y )\)\(a\)\(a\)\(a\)\(b\)\(b\)\(b\)0.110.05
    Given that \(\mathrm { E } ( Y ) = 4.02\)
  3. form and solve two equations in \(a\) and \(b\) to show that \(a = 0.15\) You must show your working.
    (Solutions relying on calculator technology are not acceptable.)
  4. Show that \(\mathrm { E } \left( Y ^ { 2 } \right) = 20.7\)
  5. Find \(\operatorname { Var } ( 5 - 2 Y )\) These dice are each rolled once. The scores on the two dice are independent.
  6. Find the probability that the sum of these two scores is 3