Random variable is a universal model of some quantity that we know with some level uncertainty. We will assume that our data are realization of some random variables. So to study data, we have to study random variables first. We'll do it using a very simple examples. First, let us consider a random experiment that we discussed previously. Coin tossing. Let us assume that we have a fair coin and we toss it to several times. Then we can consider a different random variables associated with this experiment. For example, we can count how many heads we get during this tossing or how many tails, or something else. But let us begin with the number of heads. So X is number of heads that we obtained during this tossing. Let us consider the simplest case when n equals to one. So we have only one tossing. Then we can have either zero heads or one head. We can draw the following table that relates outcome of the random experiment and values of our random variable. For one coin tossing, we have exactly two outcomes. It can be either head or tail. The corresponding value of X have to be one here and zero here. If we toss one coin and we have head, then the number of heads is one. If we toss one coin and we have tail, then the number of heads equals to zero. So this is pretty simple. Consider a little bit more interesting example. When the number of tossings equals to two. How many outcomes do we have? We have four outcomes. Just head-head. Then head-tail. Then tail-head. Finally, tail-tail. We can find the value of X for each of these possible outcomes. If the outcome is head-head, then we have two heads and the value of X is two. If we have a head-tail or tail-head, then we have on the one head during our sequence of tossing. So for both outcomes, the value of X equals to one. Finally, if in both those tossings we have tail, then we have zero heads and value of X is zero. In a similar way, we can define the corresponding number of heads for n equals to three or some other number. We can also consider non-fair coin but a coin that gives head with some probability P. This is a very well-known Bernoulli distribution and that we will discuss later. But now let us consider different random experiment and associate some random variables with it. For example, let us instead of coin tossing consider dice tossing for one dice. If we does that at one time, we have six possible outcomes. We can associate some random variable with these outcomes. For example, we can consider the square of a number of points that we get on the dice. I will denote outcomes by the corresponding pictures. Let us consider random variable Y which is the square of points. Then if we have one point, then the value of Y equals to one. If we have two points, the value of Y is four, and so on. Nine here, 16 here, 25 here, and 36 here. In these examples, the value of random variable is integer number. But in fact, it is possible to consider a random variable with non-integer values. Let us consider an example of such a random variable. Again, first we have to introduce random experiment. For example, let us assume that I have three coins. One coin is $1 coin. Another coin a half-a-dollar coin. The third coin is a quarter. Now, let us assume that my random experiment is the following. I get randomly two of these three coins, and the value of random variable is the amount of money that is in these coins. So for example, if I get an outcome that I chose these going and these coin, then the value of random variable that I will denote by Z will be a sum of these amounts. So it is one-and-a-half. Now, we can consider a different outcome that we choose for example this coin this coin. The corresponding value of Z equals to 1.25. Finally, the third possible outcome is that we chose this coin and this coin. The value of Z is 0.75. So this gives us another example of a random variable. Now, let us pause to the mathematical definition of a random variable.