[MUSIC] We often need to transform random variables, for example by adding a constant or multiplying random variable by a constant. So, it is useful to understand how probability density function changes when we do such a transform. [SOUND] First, let us consider adding a constant. Let X be random variable. And c be constant. Then we can consider a new random variable y, which is equals to X + c, and let us assume that we have probability density function for variable X. What can I say about probability density function of variable Y? Let us assume that c is some positive constant. How can we find probability density function of Y? We have to consider an event that y lies in some segment. How this event is connected with a similar event for X, let us check. If X is in some segment from x small to x small plus delta x, then it means that Y is in similar segment but shifted by this c. So it is in a segment from X + c to X + c + delta X. So when we do transformation, this segment shifts by adding this constant. And we get a new segment, As these two events occur simultaneously, their probability equals. However, the probability of this event, of getting our random variable x in this segment, is involved in the definition of the value of probability density function of f at this point. In the same way, probability of this event is involved in the definition of probability density function for random variable Y. And this is defines the value of PDF of Y, at this point X plus c. So it means that this point is shifted to this point, because this value of X corresponds to this value of Y. In the same way, the whole graph of probability density function is shifted to the right if c is positive exactly by adding plus c to all X coordinates. So, let us draw the new graph. We see that it is a shifted version of this old graph. So adding a constant giving us shifting of probability density function. What about multiplication? Let us assume that this graph is a graph of probability density function for X. What is the probability density function for Z? Again, we have to consider two events. As Z is equal to cX, this event is equivalent to event that Z is in segment cX, cX plus c delta X. This is a segment from X to X plus delta X. And when we multiply both endpoints of the segment to some constant, for example, we can think about constant like two or three. We get a new segment like this one. In contrast of the previous case, the length of the new segment is different from the length of the initial segment. So when we write that we are interested in probability density function of Z, we have to write that for example pdf of Z at point cX is equal to limit as, delta-Z tends to zero. Probability that Z is in CX, cX plus delta Z. Over delta Z. And to connect this probability with probability of this event, we have to put delta Z to equal to C delta X. Now, we can move this C out of the limit sign. We see here probability of this event and this event is the same as this event. If we replace, here, this probability by this probability we will get probability density function of X at point x small. So we see that this is equal one over c, probability density function of X at point x small. In other words, these two probability density function are related with the following transformation. First, we have to stretch the figure horizontally by multiplying X by c. And second, we have to squeeze the graph vertically by multiplying the value of the function by 1 over c. Now, the new graph looks like the following. The fact that we have to compensate expanding in horizontal direction by contraction in vertical direction preserves the probability of getting somewhere or in other sense, the area under this curve to be equal to one. So for this curve, area under the curve is equal to one because it is probability density function. And the same holds for the new curve because the expansion in horizontal direction is compensated by contraction in the vertical direction. [MUSIC]