Lesson Worksheet: Expected Values of Discrete Random Variables Mathematics

In this worksheet, we will practice calculating the expected value of a discrete random variable from a table, a graph, and a word problem.

Q1:

Work out the expected value of the random variable 𝑋 whose probability distribution is shown.

Q2:

The function in the given table is a probability function of a discrete random variable 𝑋. Find the expected value of 𝑋.

𝑥1346
𝑓(𝑥)10278𝑎6𝑎19
  • A28627
  • B7627
  • C15
  • D5827

Q3:

The frequency table shows the number of cars that 65 families have.

Number of Cars1234
Frequency1035155

Find the mean number of cars per family.

  • A2913
  • B132
  • C1329
  • D229
  • E292

This data can be expressed as a probability distribution for the discrete random variable 𝑋 as shown. Find the value of 𝑎, 𝑏, 𝑐, and 𝑑.

𝑥1234
𝑝()𝑎𝑏𝑐𝑑
  • A𝑎=213, 𝑏=713, 𝑐=913, 𝑑=413
  • B𝑎=1113, 𝑏=613, 𝑐=1013, 𝑑=1213
  • C𝑎=110, 𝑏=235, 𝑐=15, 𝑑=45
  • D𝑎=213, 𝑏=713, 𝑐=313, 𝑑=113
  • E𝑎=213, 𝑏=713, 𝑐=913, 𝑑=113

Calculate the expected value of 𝑋.

  • A2913
  • B229
  • C292
  • D1329
  • E132

Q4:

The table shows the probability distribution of a fair six-sided die. Determine 𝐸(𝑋).

𝑥123456
𝑝()161616161616

Q5:

The discrete random variable 𝑋 has the shown probability distribution.

𝑥123456
𝑝()0.10.30.20.10.1𝑘

Find the value of 𝑘.

Hence, determine the expected value of 𝑋.

Q6:

An experiment produces the discrete random variable 𝑋 that has the probability distribution shown. If a very high number of trials were carried out, what would be the likely mean of all the outcomes?

𝑥2345
𝑝(𝑥)0.10.30.20.4

Q7:

Work out the expected value of the random variable 𝑋 whose probability distribution is shown.

Q8:

Work out the expected value of the random variable 𝑋 whose probability distribution is shown.

Q9:

The function in the given table is a probability function of a discrete random variable 𝑋. Given that the expected value of 𝑋 is 4, find the values of 𝑎 and 𝑏.

𝑥13𝑏56
𝑓(𝑥)0.20.2𝑎0.20.3
  • A𝑎=0.1, 𝑏=4
  • B𝑎=0.1, 𝑏=3
  • C𝑎=0.2, 𝑏=5
  • D𝑎=0, 𝑏=3

Q10:

The function in the given table is a probability function of a discrete random variable 𝑋. Given that the expected value of 𝑋 is 25457, find the value of 𝐵.

𝑥12𝐵7
𝑓(𝑥)8𝑎3𝑎138𝑎

Q11:

Work out the expected value of the random variable 𝑋 whose probability distribution is shown.

Q12:

The function in the given table is the probability function of a discrete random variable 𝑋. Find the expected value of 𝑋.

𝑥01234
𝑓(𝑥)0.1𝑎0.10.40.2

Q13:

Let 𝑋 denote a discrete random variable which can take the values 1,𝑀,1and. Given that 𝑋 has probability distribution function 𝑓(𝑥)=𝑥+26, find the expected value of 𝑋.

  • A1
  • B13
  • C83
  • D23

Q14:

Let 𝑋 denote a discrete random variable which can take the values 1, 2, 3, 4, and 5. Given that 𝑃(𝑋=1)=733, 𝑃(𝑋=2)=833, 𝑃(𝑋=3)=111, and 𝑃(𝑋=4)=133, find the expected value of 𝑋.

  • A1211
  • B16
  • C14411
  • D10633

Q15:

Let 𝑋 denote a discrete random variable which can take the values 4, 5, 8, and 10. Given that 𝑃(𝑋=4)=427, 𝑃(𝑋=5)=527, and 𝑃(𝑋=8)=827, find the expected value of 𝑋. Give your answer to two decimal places.

Q16:

Let 𝑋 denote a discrete random variable which can take the values 2, 0, and 5. Given that the expectation of 𝑋 is 0.03 and 𝑃(𝑋=2)=925, find 𝑃(𝑋=5).

  • A320
  • B51100
  • C925
  • D1625

Q17:

The discrete random variable 𝑋 has the shown probability distribution.

𝑥1234
𝑝()𝑘1𝑘2𝑘3𝑘4

Find the value of 𝑘.

  • A110
  • B2512
  • C1225
  • D1213
  • E611

Hence, determine the expected value of 𝑋.

  • A253
  • B2411
  • C4813
  • D4825
  • E25

Q18:

A discrete random variable 𝑋 has a uniform probability distribution such that 𝑃(𝑋=𝑥)=111, where 𝑥{1,2,3,4,5,6,7,8,9,10,11}. Determine 𝐸(𝑋).

Q19:

23 students took an exam; 7 students got 3 marks, 8 students got 8 marks, and 8 students got 2 marks. Given that 𝑋 denotes the number of marks received, find the expected value of 𝑋. If necessary, round your answer to the nearest hundredth.

Q20:

In an experiment, Emma is going to spin a fair four-sided spinner numbered from 1 to 4. Chloe says that the expected value of the experiment is 2.5. Emma disagrees as she says it is impossible to spin 2.5 and suggests that the expected value is 3. Who is correct and why?

  • AChloe is correct because the expected value is the average result of an experiment after a large number of trials, which is 2.5 in this case.
  • BEmma is correct because the expected value is the average result of an experiment after a large number of trials, which is 2.5 in this case. However, this is unobtainable on the spinner, so it must be rounded to the nearest whole number, which is 3.

Q21:

In an experiment, Scarlett rolls two fair six-sided dice and adds the numbers. The probability distribution of the experiment is shown.

𝑥23456789101112
𝑝()136236𝑎436𝑏𝑐536𝑑336236136

Find the values of 𝑎, 𝑏, 𝑐, and 𝑑.

  • A𝑎=336, 𝑏=536, 𝑐=636, 𝑑=736
  • B𝑎=336, 𝑏=536, 𝑐=136, 𝑑=336
  • C𝑎=336, 𝑏=536, 𝑐=636, 𝑑=436
  • D𝑎=536, 𝑏=336, 𝑐=636, 𝑑=436
  • E𝑎=336, 𝑏=536, 𝑐=136, 𝑑=436

What is the expected value of the experiment?

Q22:

If two fair six-sided dice were rolled and the numbers were added together to form a score, the expected value would be 7. Determine which of the following statements is true.

  • AIf a 3 is rolled first, it is more likely that the second die will land on a 4 than any other number.
  • BIf a 4 is rolled first, it is more likely that the second die will land on a 3 than any other number.
  • CAfter a large number of trials, the average score would be close to 3.
  • DIf we were to roll two dice, we should not expect to ever obtain any score other than 7.
  • EAfter a large number of trials, the average score would be close to 7.

Q23:

In an experiment, Michael is going to flip four coins. How many times is it expected for him to get heads?

Q24:

Let 𝑋 be a discrete random variable with probability distribution function 𝑓(𝑥)=𝑘𝑥+46.25 and 𝑋=1,1,2,3.

Find the value of 𝑘.

Calculate the mean of 𝑋. Round your answer to the nearest hundredth.

Find 𝑃(𝑋<2).

Q25:

Two boys and two girls are ranked according to their scores on an exam. Assume that no two scores are alike and that all possible rankings are equally likely. Let 𝑋 be the random variable expressing the highest ranking achieved by a girl (e.g., 𝑋=2 if the top-ranked student is a boy and the second-ranked student is a girl). Find 𝐸(𝑋).

  • A12
  • B53
  • C6
  • D2
  • E52

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