Work out the expected value of the random variable 𝑋 whose probability distribution is shown.
This is an example of a uniform distribution. We can see that the probabilities of all possible outcomes are equal. They’re 0.2. It can be useful to represent any information in table form. It makes it easier to spot what to do next. Now, since our distribution is uniform, there is a shortcut to finding the expected value of 𝑥. But let’s first consider the formula.
To find the expected value of 𝑥, we add together each of the possible outcomes multiplied by the probability of the outcome occurring. For our distribution, the first possible outcome — the first value of 𝑥 — is one and the probability that outcome occurs is 0.2. So we write one multiplied by 0.2.
The second possible value of 𝑥 is two and the probability that that occurs is 0.2. So we write two multiplied by 0.2. The third possible value of 𝑥 is three. So we write three multiplied by 0.2. And then, we repeat that process for the remaining possible values of 𝑥, which are four and five.
Evaluating each of our products and we get 0.2 plus 0.4 plus 0.6 plus 0.8 plus one, which is three. So the expected value of 𝑥 for our probability distribution is three.
Now, it’s useful to know the formula for discrete uniform expectation. For a uniform distribution, where 𝑥 is any number from one up to 𝑛, the expected value of 𝑥 can be found by using this formula: 𝑛 plus one all divided by two. And we can use this to double-check what we’ve already done. For our distribution, 𝑥 can take any value from one to five. So 𝑛 is equal to five.
Our formula becomes five plus one divided by two or six divided by two, which is three. Once again, we’ve worked out that the expected value of 𝑥 is three.