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?
The law of large numbers, also sometimes known as the law of averages, says that the mean of the results from a very high number of trials tends to the expected value. Here that’s 𝐸 of 𝑥. As the number of trials 𝑛 tends to ∞, we say that the mean is equal to 𝐸 of 𝑥: the expected value of 𝑥. The formula we need to know for the expected value of 𝑥 is given by the sum of 𝑥 multiplied by 𝑝 of 𝑥. It’s the sum of each of the possible outcomes multiplied by the probability of this outcome occurring.
So let’s substitute what we have into this formula. 𝑥 multiplied by 𝑝 of 𝑥 for the first column is two multiplied by 0.1. For the second column, it’s three multiplied by 0.3. For the third column, four multiplied by 0.2. And for the fourth and final column, that’s five multiplied by 0.4. Two multiplied by 0.1 is 0.2, add 0.9, add 0.8, and five multiplied by 0.4 is 2.0.
So we add 2.0 at the end of this line. The sum of these values is 3.9. So the expected value of 𝑋 is 3.9. And we said that for a very high number of trials, we could call that the mean. Now we can look at our table to check whether this answer is likely to be correct. Since the possible values for 𝑥 are two, three, four, and five, and 3.9 is a little over halfway between two and five, 3.9 is likely to be correct for the mean of this probability distribution.