Video: Determining the Expected Value of a Discrete Random Variable

25 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.

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Video Transcript

25 students took an exam. Seven students got three marks, eight students got eight marks, and eight students got two 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.

A discrete random variable is a variable which can take only a finite number of values and for which the probabilities of each variable occurring sum to one. In this case, we’re told that the variable is the number of marks received. So we can take the values two, three, and eight since those are the number of marks scored by the students in the exam.

We can also work out the probability that a student picked at random scored one of these number of marks. Seven plus eight plus eight tells us that there are a total of 23 students. Eight of these students scored two marks. That means that the probability a student chosen at random scored two marks is eight over 23. Seven students achieved three marks. A probability then that a student chosen at random got three marks is seven over 23. Similarly, the number of students who got eight marks was eight. So the probability a student chosen got eight marks is eight over 23.

Now, we’re being asked to find the expected value of 𝑥. So, we need to recall the formula for this. It’s the sum of each of the possible outcomes multiplied by the probability of that outcome occurring. The first possible value of 𝑥 is two and the probability that occurs is eight over 23. So we write two multiplied by eight over 23. The second possible value of 𝑥 is three and the probability that that occurs is seven over 23. So we write three multiplied by seven over 23. And our final possible value of 𝑥 is eight and the probability that occurs is eight out of 23. So we write eight multiplied by eight over 23.

Evaluating each of these products, we get 16 over 23 plus 21 over 23 plus 64 over 23, which is equal to 101 over 23. 101 divided by 23 is 4.3913 and so on. We were told to round our answer to the nearest hundredth. That becomes 4.39.

The expected value of 𝑥 then is 4.39.

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