As we discussed before, it does not make sense to ask about continuous random variable, what is the probability that this random variable takes a particular value? However, we can ask what is the probability that our random variable takes a value which is close to some particular number? The answer for this question gives us a probability density function. Let us consider some continuous random variable. Let us consider some particular value of x-naught. How can we ask mathematically about the probability that our random variable takes the value that is close to this x-naught. We can consider some small segment that is nearer to x-naught. For example, segment from x-naught to x-naught plus Delta x where Delta x is some small number. Let us take this segment and ask what is the probability that x takes the value in this segment? As we discussed before, this probability can become very small when the length of this segment becomes small. However, we are interested in small segments. We are interested in small Delta x because we are interested in the small neighborhood of this point x-naught. So what can we do? We can divide this probability by Delta x, then as this probability decreases, this Delta x is also decreases, and it is possible that this ratio has a limit. If this limit exists, we say that this limit by definition is equal to the value of probability density function of random variable x at point x-naught. Now let us consider example that we discussed before. Let us find the probability density function for uniform distribution on segment from zero to one. We can see there are some fixed point x-naught. There are two options; either x-naught is inside of the segment or it is outside of it. Let us consider first, the case when x-naught is inside of this segment. Moreover, I demand for x-naught to be inside open interval, I exclude this borders of the segment. In this case, if we consider some small segment from x-naught to x-naught plus Delta x, this whole segment is inside of this larger segment from zero to one. In this case, probability that our random variable x is inside of this segment is equal to its length and its length is Delta x. So probability that x is inside x-naught, x-naught plus Delta x is equal to Delta x. A probability density function can be defined as a limit. As Delta x tends to zero, Delta x over Delta x. Of course, this is a limit of a constant one, and it is equal to one. This is the value of probability density function at point x-naught by definition. If we draw a graph of probability density function, we see that above this integral we have constant function which is equal to one. The second case when x-naught is outside of segment from one to zero can be treated similarly. What is the probability for our random variable to be inside some segments from x-naught to x-naught plus Delta x, if is outside of this segment. By definition our random variable takes values only on this segment. It means that the probability for our random variable to take value here is equal to zero. It means that if we consider definition of probability density function, we will also have zero. So the probability density function here and here outside of the segment is identical zero. What about these points; zero and one? It appears that probability density function is not defined at this point because this limit does not exist, but this is perfectly normal. It is okay for probability density function to have discontinuities of this kind. So far, we defined a probability density function for uniform distribution. At uniform distribution, all parts on the segment have the same probability provided that they have same length, and this can be seen by the fact that the probability density function is constant over this segment. However, we can consider different probability density functions, and we will do it later.