Together with probability density function, we need yet another one way to describe continuous random variables. This way is called to use cumulative distribution function. Let X be some random variable, and let us consider the following function. Its value from some point x is probability that our random variable X takes value that is less than or equal to x small. So basically, if we have some point here x, then we have this ray from x to negative infinity, and CDF of x is the probability that X capital is somewhere inside of this ray. Let us consider example of uniform distribution again. Let us pick some point here. What is CDF at this point? In other words, what is the probability that our random variable, which is uniformly distributed on the segment from 0 to 1? What is the probability for this random variable to take a value somewhere here on this right? It is zero. It means that for all points that to the left from the zero, the value of CDF of x at this point is equal to 0. Now, let us consider some point here. What is the probability that our random variable uniformly distributed on this segment has value that is less than, for example, some value here? Of course, if random variable is distributed only on this segment, then it necessarily to have values that are less than this value. So this happens with probability of one. It means that for all points to the right from one, the corresponding CDF is equal to 1. What about these points from here to here? We know that for every point here, let this value be x naught, the probability that our random variable is on this ray is equal to the probability that it is on this segment because there is no probability from getting here, and the probability of this segment is equal to its length, and its length is equal to x naught. So the value of CDF at this point is equal to x naught as well. Therefore, here we have a linear function. Basically, we can say that cumulative distribution function can be written in this case in the following way. It is equal to 0 if x is negative, it is equal to 1 if x is greater than 1, and it is equal to x if x is inside of segment from 0 to 1. Again, we can consider different distributions, and these different distributions will have different CDFs. We will consider this later. But now I want to say that we can define cumulative distribution function not only for continuous random variables, but also for discrete random variables that we discussed on the previous week. Let me consider some random variable. Let me assume that this variable takes value 1 with probability 0.2, takes value 2 with probability 0.5, and takes value 2.5 with probability 0.3. We can write cumulative distribution function for these new variable as well. Let us find its value and draw its graph. Our random variable can take on the values 1, 2, and 2.5. It means that if we ask what is the probability that x is less than or equal to some number that to the left of one, for example, 0.9. Of course, this probability is equal to 0 because it is not possible for this random variable to take any values that satisfy this condition. So this is equal to 0. This is true for every number that is less than 1. So here, values of cumulative distribution function R equals to 0. Now, let us consider case x equals to 1. What is probability that x is less than or equal to 1? We see that x cannot be less than 1, but it can be equals to 1. So this probability is actually equal to probability that x equals to 1, and this probability is equals to 0.2. This is the value of cumulative distribution function at 0.1. Now let us choose any point between 1 and 2. Again, if we ask what is the probability, for example, that x is less than or equal to 1.5? The only possible way for x to satisfy this condition is to be equal to 1. So this probability again equals to probability of event that x equals to 1. So it is again equals to 0.2, and this is value of CDF at point 1.5. The same true for any point between 1 and 2, except 2. So we can draw the graph in this way. Now let us ask, what is the probability for x to be less than or equal to 2? There are two cases for this event, x can be either equal to 1 or equal to 2. So this probability basically equals to probability that x equals to 1 plus probability that x equals to 2. So we have to sum these two values. This is the value of cumulative distribution function at point 2. In the same way, for any point between 2 and 2.5, the value of CDF is equals to the same value 0.7, except point 2.5. Now, we can see there point 2.5 or any point that is larger than this one and ask a similar question, what is the answer? We know that x can take value only in this set, and every element of this set is less than or equal to 2.5. So the probability for larger points is equal to 1. This is typical cumulative distribution function for discrete random variable. It has several jumps, and each jump corresponds to a particular value that our variable can take the size of this jump. For example, this difference is equal to probability of the corresponding value. For example, the difference between this and this is equal to 0.5, and this is this probability. So we see that cumulative distribution function can define random variables that are continuous and that are discrete. It is a kind of universal tool describe random variables. Let us discuss some properties of cumulative distribution function.