Now, we are ready to consider some important examples of random variables that are defined by their probability density functions. Previously, we discussed only a uniform distribution on segment from zero to one, but it is possible to define similar distribution on any segment. Let us find its probability density function. As previously, outside of this segment, the probability density function have to be equal to 0 because no points here or here are possible. As we demand our distribution to be uniform on this segment, the value of probability density function over this segment should be constant. The rest is to find this constant. Note that probability density function have to have the following property. Integral of probability density function over all line have to be equal to 1. Indeed, this integral represents probability that our random variable takes some value between minus infinity and plus infinity. But in any case, random variable takes a value between minus infinity and plus infinity. So the probability of this event is equal to 1. So this integral have to be equal to 1. In this case, outside of the segment from a to b, the value of p of x is equal to 0. So instead of this integral from minus infinity to plus infinity, we can consider integral from a to b. So we have integral from a to b, and the value of probability density function is a constant and it is equal to P naught. So we have this relation. When we integrate constant, we can just move it out of the integral and it is easy to see that this integral is just an area of a rectangle for which one length is equal to b minus a, and the other is equal to P naught. So we have b minus a P naught equals to 1, and it follows immediately that P naught is equal to 1 over b minus a. Now, we can start at uniform distribution on arbitrary segment from a to b. Probability distribution function is equal to 0 if x is outside of a, b and it is equal to 1 over b minus a for x inside interval a, b. Again, this function is not defined at the end points of the segment, but this is completely normal. Now, we can consider a non-uniform distribution on some segment. Let us begin with segment from zero to one. Let me assume that I want the distribution that gives higher probability to get a point near zero than near one. In this case, I can choose a different function as probability density function. Not a constant over the segment, but some other function that is large here and small here. For simplicity, let us choose linear function. Again, we have a question. What is the value here? Again, this can be found by the same condition as previously. Integral from 0 to 1 of p of x dx should be equal to 1. This integral is an area of triangular and this area, by well-known formula, is equal to this side 1 times this side P naught times 1/2 because it is an area of triangular and not a rectangle. So we see that P naught should be equal to 2 in this case. This is an example of non-uniform distribution on a segment. If we generate points from this distribution, meaning that we do the random experiment and we get a value of a random variable and we do it several times, we will get some points. Sometimes we will get points which are far from origin, but points near the origin will be obtained much more frequently than points far from the origin, and this is what this probability density function says about this distribution. Finally, let us discuss an example of a random variable that take a value not on some segment, but on the whole line. This is a very well-known and a crucial distribution which is called normal or Gaussian distribution. We will consider so-called standard normal distribution. Standard normal distribution has a PDF that which graph is a well-known bell-shaped curve. It looks like this one. So theoretically, for normally distributed random variable, it is possible to have any value on the line. However, the most of the values, the most of probability density is not far from the origin. Mathematically, probability density function for normal distribution is given by the following formula. We see that as x increases, this exponent becomes very, very, very small and we see the same thing on the graph. Normal distribution is a very important example of distribution, and we will meet with this distribution later when we will discuss central limit theorem.