Work out the expected value of the random variable 𝑋 whose probability distribution is shown.
A probability distribution is usually given as a table of values showing the probabilities of various outcomes of an experiment. However, the information here is represented as a bar chart. It would be hugely useful then to convert this information into table form.
We can see that our values of 𝑋 are one, two, three, four, and five. And their respective probabilities are given by the height of each bar. One has a probability of 0.1, two has a probability of 0.2, three a probability of 0.4, four a probability of 0.2, and five a probability of 0.1.
We can check if we’ve analyzed our bar chart correctly by checking that all of our probabilities sum to one. 0.1 add 0.2 add 0.4 add 0.2 add 0.1 does indeed equal one. In this case then, all we need to do is apply the formula for the expected value of 𝑋, also known as the mean.
That formula may look a little bit scary. But remember, all it means is the sum of the probabilities multiplied by their random variable. One multiplied by 0.1 add two multiplied by 0.2 add three multiplied by 0.4 add four multiplied by 0.2 add five multiplied by 0.1 is three. The expected value of the random variable 𝑋 whose probability distribution is shown is therefore three.