Let us discuss symmetric distributions. It means that probability mass function have some vertical axis of symmetry. Let me show you a graph to make it clear. Let us assume that we have a probability Mass function that looks like something like this. We have axis of symmetry. Which is vertical axis like X = x0. For each value like this, we have symmetric value like this and our random variable takes this and this values with equal probabilities. So the overall graph is symmetric with respect to this vertical line. In this case, it appears that we can find expected value of the corresponding random variable without any calculations. In fact the expected value of X = x0, at least if it exists, it exists anyway, if we have only finite number of values, but for infinite numbers, it can be not true, but we consider only case if this expected value exists. Let us prove this fact using our properties of expected value that we discussed. First of all, let us consider new variable Y which is equal to X- x0. Probability Mass function for variable Y is related to probability Mass function of X. In fact, we just shift this graph to the left in such a way that this vertical line becomes a coordinate axis for this new random variable. So if we denote this axis as axis for probability Mass function of variable X, then we can introduce new vertical axis, which is our probability Mass function of variable y, then probability Mass function of Y is symmetric with respect to the vertical axis. It means that this function is even, the symmetry of this distribution means that the distribution of Y is the same as the distribution of negative y, it means that they have the same expected value. So expected value of Y is equal to expected value of negative Y. Now let us apply rules that we discussed, negative Y is the same as negative 1 times Y, it means that this is the same as negative 1 times Y and we can move this negative 1 out of the expected value sign. And in this case, we see that it is equal to negative 1 times expected value of Y. So we see that expected value of Y is a number that does not change when we multiply it by negative 1, the only possible value for expected value of Y and thus is to be 0. Now we can return to our variable X, expected value of Y is equal to expected value of X minus x0. It means that it is equal to expected value of X minus x0, according to the first rule that we discussed above. And according to this outcome, we see that this thing is equal to 0, it means that expected value of x is equal to X0 basis that we announced and wanted to prove, so we finish this proof. Now, let us return to probability Mass function and discuss, why does it work in in this way? It is quite intuitive indeed. We have this value X0 and the value of our random variable X can deviate from X0 either in positive or in negative direction, but these deviations in positive and in negative directions are the same. They have the same probabilities and it means that when we make a summation they will cancel each other. This is why this value X0 is the actual expected value of x, this gives us some intuition about expected values and how they work. [MUSIC]