Random variables can represent different things. For example, some random variables can represent something like number of heads in a series of coin tossing. This number can take on the finite or countable many values. However, it is possible that we need random variable that measures something like temperature or height or some other number that can take any values in some segment or at the whole real line. Now we will discuss random variables of this second kind, random variables that can take any value from some segment for example. But to begin, let me recall how we classify random variables. Previously, we discussed discrete random variables. We also subdivide discrete random variables into two sub-categories, random variables that can take only finite number of values. In this case, to specify a random variable, we specify its distribution, meaning that we specify which values this random variable can take and with which probabilities. So we have X_1 and so on, X_n. This is values and P_1 and so on, P_n probabilities. Another option for discrete random variable is to take infinite many but countable number of values. In this case, we have an infinite sequence of values, X_1, X_2 and so on, X_n and so on, and infinite sequence of probabilities. Now we'll discuss a new kind of random variables, which are called continuous random variable. Continuous means that random variable can take any possible value, for example, in some segment or at the whole line. Now we will discuss how to define a continuous random variables. Let us assume that we want to pick a random number from one to zero. Geometrical speaking, we have a segment, and we want to pick a random point on this segment. We have a segment from one to zero, and our random variable is some random point in this segment. Let us also assume that we want to choose this point uniformly over this segment. It means that, for example, the probability for this point to be at the left half on the segment is the same as the probability of this point to be at the right half on the segment. There is no preference between left and right parts. If we said that we need this point to be inside this segment, then the probability that this point is in this segment is equal to one. What is the probability that this point is in the left half of the segment? As we discussed, this probability have to be equal to this probability. However, we see that the probability of that point is here and probability that this point is here in some should gives us the probability that this point is somewhere on this segment. It means that the sum of these two probabilities have to be equal to one. Then both of them should be equal to one-half. In the same way, we can split our segment into four parts. The probability of the event that our random variable is in this part is the same as in this part and is the same as in this part and in the same as in this part. It means that all these probabilities have to be equal to one-forth. Using this kind of reasoning, we can decide that for any segment, like from a to b, the probability of this point to lie inside this segment is equal to the length of the segment, which is quite natural. The larger segment we consider, the higher probability that our random variable lie inside this segment. However, we have a problem. Let us ask, what is the probability that our random variable is equal to some predefined point, some point X naught? We see that we can take arbitrary small segment that contains this point. The probability that X is equal to X naught is not larger than the probability that X is inside segment from X naught minus Epsilon to X naught plus Epsilon. But this probability is equal to the length of this segment. It is two Epsilons. So we see that this probability have to be less than two Epsilons, and this holds for every positive Epsilon. It means that this probability have to be equal to zero. We see the problem. Previously, we described random variables in terms of their possible values and the probabilities, that random variable, and take this particular value. But in case of continuous random variables, we cannot do it like this because the probability that random variable take any particular value is equal to zero. Now we see that we cannot describe continuous random variables in terms of the probability of their values as we did it for discrete random variables. However, we can use a different notion, a notion of probability density function, to describe continuous random variables. Just a little note, here is not the full description of all possible types of random variables. We will study only discrete and continuous random variables. But mathematically speaking, it is possible to consider more complicated random variables. One example of such random variable will be discussed at the end of this week's lectures. If you want to study the full possibilities of random variables, you have to study some measure theory. This is outside of the scope of this course.
