# Question Video: Calculating Expected Values Mathematics

Work out the expected value of the random variable π whose probability distribution is shown.

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### Video Transcript

Work out the expected value of the random variable π 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. Theyβ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 π₯. But letβs first consider the formula.

To find the expected value of π₯, we add together each of the possible outcomes multiplied by the probability of the outcome occurring. For our distribution, the first possible outcome β the first value of π₯ β is one and the probability that outcome occurs is 0.2. So we write one multiplied by 0.2.

The second possible value of π₯ is two and the probability that that occurs is 0.2. So we write two multiplied by 0.2. The third possible value of π₯ is three. So we write three multiplied by 0.2. And then, we repeat that process for the remaining possible values of π₯, 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 π₯ for our probability distribution is three.

Now, itβs useful to know the formula for discrete uniform expectation. For a uniform distribution, where π₯ is any number from one up to π, the expected value of π₯ can be found by using this formula: π plus one all divided by two. And we can use this to double-check what weβve already done. For our distribution, π₯ can take any value from one to five. So π is equal to five.

Our formula becomes five plus one divided by two or six divided by two, which is three. Once again, weβve worked out that the expected value of π₯ is three.