# 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 whose probability distribution is shown. Q2:

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

 𝑥 𝑓(𝑥) 1 3 4 6 1027 8𝑎 6𝑎 19
• A
• B
• C15
• D

Q3:

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

 Number of Cars Frequency 1 2 3 4 10 35 15 5

Find the mean number of cars per family.

• A
• B
• C
• D
• E

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

 𝑥 𝑝() 1 2 3 4 𝑎 𝑏 𝑐 𝑑
• A, , ,
• B, , ,
• C, , ,
• D, , ,
• E, , ,

Calculate the expected value of .

• A
• B
• C
• D
• E

Q4:

The table shows the probability distribution of a fair six-sided die. Determine .

 𝑥 𝑝() 1 2 3 4 5 6 16 16 16 16 16 16

Q5:

The discrete random variable has the shown probability distribution.

 𝑥 𝑝() 1 2 3 4 5 6 0.1 0.3 0.2 0.1 0.1 𝑘

Find the value of .

Hence, determine the expected value of .

Q6:

An experiment produces the discrete random variable 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?

 𝑥 𝑝(𝑥) 2 3 4 5 0.1 0.3 0.2 0.4

Q7:

Work out the expected value of the random variable whose probability distribution is shown. Q8:

Work out the expected value of the random variable whose probability distribution is shown. Q9:

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

 𝑥 𝑓(𝑥) 1 3 𝑏 5 6 0.2 0.2 𝑎 0.2 0.3
• A,
• B,
• C,
• D,

Q10:

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

 𝑥 𝑓(𝑥) 1 2 𝐵 7 8𝑎 3𝑎 13 8𝑎

Q11:

Work out the expected value of the random variable whose probability distribution is shown. Q12:

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

 𝑥 𝑓(𝑥) 0 1 2 3 4 0.1 𝑎 0.1 0.4 0.2

Q13:

Let denote a discrete random variable which can take the values . Given that has probability distribution function , find the expected value of .

• A1
• B
• C
• D

Q14:

Let denote a discrete random variable which can take the values 1, 2, 3, 4, and 5. Given that , , , and , find the expected value of .

• A
• B16
• C
• D

Q15:

Let denote a discrete random variable which can take the values 4, 5, 8, and 10. Given that , , and , find the expected value of . Give your answer to two decimal places.

Q16:

Let denote a discrete random variable which can take the values , 0, and 5. Given that the expectation of is 0.03 and , find .

• A
• B
• C
• D

Q17:

The discrete random variable has the shown probability distribution.

 𝑥 𝑝() 1 2 3 4 𝑘1 𝑘2 𝑘3 𝑘4

Find the value of .

• A
• B
• C
• D
• E

Hence, determine the expected value of .

• A
• B
• C
• D
• E

Q18:

A discrete random variable has a uniform probability distribution such that , where . Determine .

Q19:

23 students took an exam; 7 students got 3 marks, 8 students got 8 marks, and 8 students got 2 marks. Given that denotes the number of marks received, find the expected value of . 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.

 𝑥 𝑝() 2 3 4 5 6 7 8 9 10 11 12 136 236 𝑎 436 𝑏 𝑐 536 𝑑 336 236 136

Find the values of , , , and .

• A, , ,
• B, , ,
• C, , ,
• D, , ,
• E, , ,

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 be a discrete random variable with probability distribution function and .

Find the value of .

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

Find .

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 be the random variable expressing the highest ranking achieved by a girl (e.g., if the top-ranked student is a boy and the second-ranked student is a girl). Find .

• A
• B
• C6
• D2
• E