Worksheet: Expected Values of Discrete Random Variables

In this worksheet, we will practice calculating the expected value from both a table and a graph and calculating the variance for a probability distribution.

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 value of 𝑎.

𝑥01234
𝑓(𝑥)2𝑎0.30.3𝑎𝑎

Q3:

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

Q4:

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

Q5:

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

𝑥123456
𝑝()161616161616

Q6:

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.

Q7:

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

Q8:

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

Q9:

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

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 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

Q11:

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

𝑥2345
𝑓(𝑥)7𝑎5𝑎9𝑎3𝑎
  • A0
  • B17
  • C124
  • D112

Q12:

An experiment that produces the discrete random variable 𝑋 has the probability distribution shown.

𝑥2345
𝑝(𝑥)0.10.30.20.4

Calculate 𝐸(𝑋).

Calculate 𝐸𝑋.

The variance of 𝑋 can be calculated using the formula Var(𝑋)=𝐸𝑋𝐸(𝑋). Calculate Var(𝑋) to 2 decimal places.

Q13:

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

Q14:

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

Q15:

Scarlett had a spinner with ten equal sections labeled with the numbers 1 to 10. She spun it 300 times and recorded the outcomes in a frequency table.

Number12345678910
Frequency35272211242833354936

If the spinner was fair, how many times would you expect to see each number if you spun it 300 times?

State whether the spinner is biased and why.

  • AThe spinner is not biased because most of the numbers appeared around 30 times.
  • BThe spinner is biased because the number 4 only appeared half as often as expected and the number 9 appeared much more often than expected.
  • CThe spinner is biased because the numbers did not appear exactly 30 times each.

Q16:

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

Q17:

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

Q18:

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

Q19:

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.

Q20:

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

Q21:

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

Q22:

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).

Q23:

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.

Q24:

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.

Q25:

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

𝑥23456789101112
𝑝()136236𝑎436𝑏𝑐536𝑑336236136

Find the value 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?

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