Today, we will start the systems of random variables. What do we mean under system of random variable? Let us consider some random experiment. We know that to define random experiment, we have to specify probability space that is associated with it. It means that we have to define Omega, which is set of elementary outcomes, and we have to define probability of every outcome. Now let us consider two random variables associated with this random experiment. Mathematically speaking, it means that we have to define two functions from Omega to space of all real numbers. Now we can say that X and Y are system of random variables. What does it mean that it is a system? These two variables can interact with each other. It means, for example, that if we know something about X, we can say something about Y and vice versa. So today, we will study these interactions between such variables. Let us discuss how can we create new random variable from existing random variable. So we have X and we want to define Y using this variable X. This can be done in different ways, and let us discuss first some important examples. Let X be some random variable and let c be some constant. So c is just a number like two. Now we can consider new random variable Y which is equal to X plus c. Let us consider example. Let me assume that I have the following probabilities; 0.2, 0.5, 0.3, and let me assume that X takes value minus 1 with probability 0.2, 3 with probability 0.5, and 4 with probability 0.3. Then we can consider a variable Y which is equal to X plus 2. It means that with probability 0.2, it takes value 1; with probability 0.5, it takes value 5; and with probability 0.3, it takes value 6. So we have new random variable that is connected with the initial variable X. It may be useful to think about probability mass functions that are associated with this transformation. Indeed, let us consider a probability mass function for variable X. Let us draw the graph of this probability mass function. So we have X here and we have negative one, and we have three here, and we have four here, and here we have a point here, here, and here. This is probability mass function for X. When we consider probability mass function for Y, we have to add to X coordinate of every point here this constant c. In this case, two. It means that all these points will be shifted to the right by two units. We see that ending of constant shifts probability mass function to the right or to the left. Another natural transformation that can we do with the random number is multiplication by a constant. Again, let c be a constant like two. Now we can consider a new random variable Z, which is equal to c times X. Now we see that if X takes a value negative 1, then Z takes value negative 2. If X takes value 3, then Z takes value 6 and so on. So we have new random variable that takes a value negative 2, 6, and 8 with these probabilities. Again, we can think about probability mass functions. Let us draw the graphs. Now, the horizontal coordinate of every point is multiplied by two. It means that the whole picture stretches by two times. So we have new numbers here. This point goes here, this point goes here, and these points go here. So this is a probability mass function which is associated with new random variable Z. Now, let us discuss how expected values change when we do these kind of transformations.