# Video: Calculating Expected Values

A discrete random variable π has a uniform probability distribution such that π(π = π₯) = 1/11, where π₯ β {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11}. Determine πΈ(π).

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### Video Transcript

A discrete random variable π has a uniform probability distribution such that the probability that π is equal to π₯ is one eleventh, where π₯ is an element of the set containing the numbers one, two, three, four, five, six, seven, eight, nine, 10, 11. Determine the expected value of π.

Notice how each outcome has an equal probability of occurring. This is an example then of a uniform distribution. There is a shortcut to help us find πΈ of π for uniform distribution. But first, weβll consider the general formula for the expected value of π₯.

The expected value of π is found by adding together all of the possible outcomes multiplied by the probability of that outcome occurring. In the case of our probability distribution, the first possible value of π, the first outcome, is one and the probability of that occurring is one eleventh. So we write one multiplied by one eleventh.

The second possible value of π is two, and the probability of that occurring is once again one eleventh. So we write two multiplied by one eleventh. The next possible value of π is three, so we repeat this process for three, and then for all the remaining possible values of π.

Remember, each of these values has a probability of occurring of one eleventh. And if we evaluate the sum of these products, we get that the expected value of π is six.

Now we did say that thereβs a shortcut to help us find the expected value of π for a uniform distribution. For a uniform distribution, where π₯ can be any number from one through to π, the expected value of π is given by π plus one over two. For our distribution, π₯ was any number from one through to 11. So π becomes 11, and our expected value of π becomes 11 plus one over two, which is once again six.