Calculate E(X) from constructed distribution

The question requires first constructing or deriving the probability distribution from a scenario (dice, games, matching problems) before calculating E(X).

4 questions · Moderate -0.8

5.02b Expectation and variance: discrete random variables
Sort by: Default | Easiest first | Hardest first
CAIE S1 2016 November Q2
5 marks Moderate -0.8
2 Two fair six-sided dice with faces numbered 1, 2, 3, 4, 5, 6 are thrown and the two scores are noted. The difference between the two scores is defined as follows.
  • If the scores are equal the difference is zero.
  • If the scores are not equal the difference is the larger score minus the smaller score.
Find the expectation of the difference between the two scores.
OCR MEI S1 2006 January Q2
8 marks Moderate -0.8
2 Four letters are taken out of their envelopes for signing. Unfortunately they are replaced randomly, one in each envelope. The probability distribution for the number of letters, \(X\), which are now in the correct envelope is given in the following table.
\(r\)01234
\(\mathrm { P } ( X = r )\)\(\frac { 3 } { 8 }\)\(\frac { 1 } { 3 }\)\(\frac { 1 } { 4 }\)0\(\frac { 1 } { 24 }\)
  1. Explain why the case \(X = 3\) is impossible.
  2. Explain why \(\mathrm { P } ( X = 4 ) = \frac { 1 } { 24 }\).
  3. Calculate \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).
AQA Further Paper 3 Statistics 2021 June Q3
5 marks Moderate -0.5
3 In a game, it is only possible to score 10, 20 or 30 points. The probability of scoring 20 points is twice the probability of scoring 30 points.
The probability of scoring 20 points is half the probability of scoring 10 points.
3
  1. Find the mean points scored when the game is played once, giving your answer to two decimal places.
    3
  2. Mina plays the game.
    Her father, Michael, tells her that he will multiply her score by 5 and then subtract 10 He will then give her the value he has calculated in pence rounded to the nearest penny. Calculate the expected value in pence that Mina receives.
OCR MEI S1 Q2
Easy -1.2
2 Every day, George attempts the quiz in a national newspaper. The quiz always consists of 7 questions. In the first 25 days of January, the numbers of questions George answers correctly each day are summarised in the table below.
Number correct123
Frequency123
  1. Draw a vertical line chart to illustrate the data.
  2. State the type of skewness shown by your diagram.
  3. Calculate the mean and the mean squared deviation of the data.
  4. How many correct answers would George need to average over the next 6 days if he is to achieve an average of 5 correct answers for all 31 days of January?