Continuing our discussion of probability distributions, now I want to look at expected values. So, here are the extracts from the reference handbook and the expected value of a discreet variable, x, with a probability mass function f of x is given by this expression. MU is equal to the expected value of x is equal to the summation of k equals 1 to n, x k times f of x k. So mu here, we use for the expected value, not to be confused with MU that we previously used for the mean of a population. So basically, what the expected value is, is the probability weighted average of all possible values. In other words, it's an average of all the values which are weighted by the individual probabilities of those individual values occurring. The variance is defined as here, sigma squared is V(X) is the summation from k equals 1 to n of x k minus the expected value, squared time f of x k. And what this is, is a measure of the dispersion of the range of possible values around the expected value. Again, we also have the continuous variable to account for here, and the definition is given in the reference handbook, but I won't cover that here because I don't think that questions about continuing with some variables are very likely. Let's do an example. So, we have newborn babies are rated on so-called Apgar scale, which ranges from zero to ten. So, zero would be a baby in very poor health, up to the maximum ten, which is a baby in very good health. And at a certain hospital, it's observed that the probability mass function of the scale is as shown here. What is the expected value of the score? So for example, here we have the number that have an x of 0 is 0.1%, the number that have 1 is 0.2%, etc., up to the number which have 10 is about 1%. So, which of these alternatives, is the expected value of the score for this particular hospital? So, here's our definition then, the expected value is the weighted average of all of the values. So, in this case the equation looks like this. So we have zero times .01 plus one times, 0.02 etc, all the way up to the last value. So computing that out, the numerical value is 6.969, but of course, this must actually be an integer value, so the answer is C. And this concludes my discussion of expected values.