Cumulative distribution function is mathematically convenient way to describe random variables. However, we are more interested in the probability density function because it is easier to interpret them. So it is important to discuss the relation between cumulative distribution function and probability density function. This relation is mathematically very nice. Let X be some random variable. Let X be a random variable for which probability density function is defined. Let us denote CDF x as F, and let us denote probability density function of x as p of x. Then let us consider some segment from a-b. Probability that X takes value in this segment can be found as a difference between values of F at point b, and at point a. What is the value of F of B? It is the probability that X takes value somewhere on this re. What is the value only F of a? It is the probability that X takes value somewhere here. So the difference between this probability and this probability is equal to probability of getting somewhere here. Technically speaking, this is the probability not of this segment but half open interval. Technically speaking, we have to put round bracket here. But as the probability of individual point is equal to 0, it doesn't change anything. So we can replace round bracket with square bracket here. Now, we can find probability density function in terms of cumulative distribution function. To do so, let us recall the definition of probability density function. Now, let us replace this probability by the difference between values of cumulative distribution function in the end points of this segment. We will have this expression is a very well-known from the cars of calculus and gives us the derivative of function F. Here is Delta F, and here is Delta x. So this is exactly a derivative F prime at point x. So we see that probability density function is a derivative of cumulative distribution function. This allows us to use PDF to reconstruct CDF. If PDF is derivative of CDF, then CDF is antiderivitive of PDF, and its values can be found by integration. Now, let us recall fundamental theorem of calculus applied to our functions. It gives us the following. Integral over segment from a to b of F prime of x dx is equal to difference between F of b and F of a. Here, we have F prime which is equal to probability density function. So we can rewrite this integral in the following way. It is integral from a to b, P of x dx. This follows from fundamental theorem of calculus. We can provide some probabilistic interpretation of this formula. We know that this difference equals to the probability that random variable takes a value on the segment from a to b. So if we have some probability density function, and we have some segment from a to b, and we asked ourselves what is the probability for our unknown value to take a value inside this segment? This formula gives us the answer. This probability that x is inside a, b equals to the difference between Fb and Fa. According to this formula, it is equal to integral of probability density function. So the integral is an area of a figure that lies under the graph of probability density function. In fact, this is quite natural. In those points for which probability density function is large, the probability to get value of random variable near this point is higher than to get value that is in the parts of the line for which probability density function is small. So it is more likely to get point somewhere here than somewhere here when we generate our random variable. More formally, we can say it in the following way. Let us consider some small segment. The probability that our random variable takes value in this segment can be calculated in the following way. From this relation, we can see that this probability is approximately equal to product of this probability density function and Delta x. So geometrically, if this is Delta x, then geometrically, this probability is equal to area of some rectangle like this one. If we split the segment into small segments and construct a rectangle like this over each of the segment, we will have a figure that approximates this figure under this graph, and the sum of these areas will give us approximately area of this figure which is this integral. So we see that we can use integrals to find probabilities and that random variable takes values in some segment. This allows us to give a good visual interpretation of probability density functions.