In this subsection, I would like to show a couple of properties of the infinitely divisible distributions and to prove that the Bernoulli distribution and the uniform distribution are not infinitely divisible. Let me formulate a couple of properties of an infinitely divisible distribution. The first property says, the characteristic function of any infinitely divisible distribution is not equal to zero for any real U. In other words, this equation a phi of U is equal to zero, does not have any real solutions. This fact is rather interesting because you know that phi is a complex-valued function, and if phi is a polynomial of order N, then it should have a root, the equation should have a root. And here, from this fact, it follows that all of these roots are complex, not real. Why this fact helps to show that a uniform distribution is not infinitely divisible? Well, let me calculate the characteristic function of uniform distribution. Let me assume that xi is uniform on the interval a_b. And in this case, the characteristic function of xi is equal to the integral from a to b, exponent in the power iux multiplied by 1 divided by b minus a, dx. And this integral is equal to 1 divided by b minus a, exponent in the power complex unit ua divided by iu, and multiply it by exponent is the power iu, b minus a minus 1. Here, I assume that u is not equal to zero. Otherwise, the characteristic function is simply equal to 1. Now, I would like to show that the equation that phi xi of U is equal to zero has some roots. Okay, this expression is equal to zero, if and only if, exponent is the power iu b minus a minus 1 is equal to zero. And this means that U is equal to 2 pi K divided by b minus a, where K is an integer number and does not equal to zero, because the case U equal to zero we have considered separately. So, we conclude that the equation that phi of U is equal to zero, has some roots and therefore, uniform distribution cannot be infinitely divisible. Okay, the second property, which is also very interesting and which holds for any infinitely divisible distribution, is that the support of a random variable have an infinitely divisible distribution is unbounded. What does it mean? So, support of a random variable is by definition a support of its probability distribution. Probability distribution of Borel subset B is a probability that this random variable is from B. And support of any measure, by definition, is a set of all X's such that for any open set G containing X, the measure of this G is strongly positive. Let me consider the support of Bernoulli random variable. You know that Bernoulli random variable xi is equal to 1 with some probability P and is equal to zero with probability 1 minus P. Therefore, we have the following pictures. There are two points, zero and one, and both of these points are definitely in the support of xi. Just look at this definition. If we take any set containing zero one, for instance, a set containing one, the probability is that xi is in this set is stronger, larger than zero, because the probability that xi is equal to zero is equal to 1 minus B. The same situation is with all sets containing 1. So definitely, zero and one, these two points, are in the support of the probability measure of xi. Otherwise, it would take any point which is not equal to zero and 1, this point is not in the support because if you consider some namely any small set containing this point, the probability that xi is the support is equal to zero. So, we conclude that the support of xi is equal to two points, zero and one, and if xi will be an infinitely divisible distribution, the support of xi should be unbounded. And therefore, we conclude that the distribution of xi is not infinitely divisible. So, now we'll have a couple of concrete examples of infinitely divisible distributions and we have a couple of interesting properties of any infinitely divisible distribution. As end, let me mention that most distributions are infinitely divisible and actually, these two examples are the only examples of not infinitely divisible distribution which are widely used in the probability theory.