Welcome to our next module of an intuitive introduction to probability, decision making in an uncertain world. The title of our new module is discreet random variables and probability distributions. So, let's dive right in. The concept of the name "Random Variable" often confuses people. What is a random variable? It's actually a very natural concept that I am sure you have used before. So, let's start with an example. We call the experiment of rolling two fair dice. We had an example a while ago and what are the possible basic outcomes? There are all the possible pairs between the number on the first die and the number on the second die. So, one-one, two-two, all the way to six-six. We have the thirty six pairs of two numbers and they are all equally likely. Often however, we're not really interested in the pair but we're interested in the sum of the pair if you play a game. And so then, what do we do? We just add the two numbers, one-one, that is a two one-two and two-one are three. we have seen this before. And so what we're doing actually, while building the sum we give to each basic outcomes a numerical value. Et voilà ! That's already a random variable. Here's the technical definition. A random variable assigns a single number a single numerical value to every basic outcome. Now unfortunately, I have to introduce just a little notation. I usually I try to avoid notation because I don't want to confuse the students with notation but here we need just a tiny little bit of notation. Usually, random variables need a name and unless we have a very good name from the context of a business application or some other real world application we use capital letters: X, Y and Z Now, a discreet random variable is a random variable where we only have finitely many possible values or perhaps infinitely many values, but values I can count. But, for now for the sake of this module focus on the list of possible values. So with the sum of two dice, the possible values would be two. three, four, all the way to twelve. Now we want to talk about probabilities. Here, with a random variable we call the collection of the probabilities the probability distribution. It gives the probability to every possible value the random variable can take on. Again, a little notation. What's a probability we write P of x some like to write Pr for the probability of x some people like to write P of X equal x that may look a little funny to you but what that means is the probability that the random variable X takes on a specific value and that's been ordered by the x. I don't care which notation you use, but if you look at textbooks if you surf the internet, you will most likely find one of these four ways to write down a probability. The next step then is, sometimes we sum probabilities and that gets us to the concept of the cumulative probability distribution. What the heck is that? Now, if I think of the cumulative probability of a number of a random variable I want to get all the probabilities for numbers less than or equal to the number of which I'm looking at. Sounds a little complicated. Here, the notation would be probability that the random variable capital X is small-equal particular value little x. Let's look at some example, and then I think it would quickly get very clear. Back to our two dice. We have seen before the probability of a two that's just a pair of one-one, only one out of thirty six pairs that gives me the sum of two, would be one in thirty six, 1/36. Then the probability that we roll the sum of three that's a couple, one-two or the other way around the couple two-one, so there are two possibilities out of thirty six probability 2/36. and here you see a list all the way to the number twelve. Now notice, we cannot roll an 11.2. I cannot roll a Pi, a zero, or minus five. All those numbers have a probability zero. Now, let's return to the cumulative probability. Here, from a spreadsheet I've put together a table that shows us first in the first column, the eleven possible values that a sum can take, two, three, all the way to twelve in the next column, the individual probabilities 1/36 for two, 2/36 for three, 6/36 for seven the most likely value, and then it goes down again all the way to 1/36 for the number twelve. Finally in the last column we have the cumulative probability. What is the cumulative probability, for example, of a four? That's a probability that if we get a two or a three or a four so you see F of x, therefore four is 0.16666, and so on. That's actually the sum of the three individual probabilities. Here's an easy rule, for the largest number the F of x is always one. In this case, the largest number possible is a twelve so for sure, with probability of one we will roll a number of twelve or less. Now, often people don't like these long excel tables too hey get to cumbersome, all these numbers these many digits, so many people prefer a graphical representation of these probabilities. So, here we have a graphical representation of the individual probabilities. Again, we see all the numbers, two to twelve and those displayed here on the horizontal axis, the x axis on the vertical axis, the y axis we see the level of the probabilities. You see, the largest bar is at the number seven and it goes all the way to 0.16 because the seven is the most likely number when you roll two dice. You see then a nice symmetric form with the smallest probabilities for two and the twelve. So these graphs of probability distribution are very popular. Occasionally you will also find a graph of the cumulative distribution. Here again, we go up in ur numbers from two to twelve and we see that we get this type of step function that's very typical for the graph of the cumulative distribution of a discreet random value and we always see the steps as we hit the number three four, five, all the way to twelve. Now technically, I can also ask what's the probability I get the number of Pi or smaller. I can look at this, Pi is between three and four so the bar in the graph shows me what the cumulative probability of that number is. Any number smaller than the smallest number possible, the two has the cumulative probability of zero and anything twelve or larger gets a number one. Now, let me show you a different example that many of you may be familiar with and that's the game of roulette. So, what in roulette happens is that there's a little ball that rolls around the wheel and eventually falls into a so called pocket. every pocket is connected to a particular number. Here in the picture I have the wheel as it's used in Europe with thirty seven possible numbers that's zero and the numbers one through thirty six. In the United States, for example in Las Vegas there are actually thirty eight pockets there's a zero and a double zero. Now let's take this game of roulette and sort of describe it in the words of probability. Here we have a random variable, the possible values here I call the random variable R of roulette and it has the possible values zero, one two, three, all the way to thirty six. All numbers are equally likely. Quick, aside some pros may say: "The croupier may actually be able to game the ball a little bit and speed it up a little faster, or little more slowly to aim for a particular pocket." We don't think of that. We assume an absolutely fair game. That's the side for the pros among you. So here, now we assume that all 37 numbers are equally likely. This probability distribution is also called the uniformed distribution Here, I wrote it down for you we get the numbers from zero, one, two, all the way to thirty six and they all have a probability of 1/37. Remember the 37th number is zero. And now we have the graphical representation of it all numbers are equally likely. This graph looks pretty boring to me. It also gets convoluted, we have now 37 of these bars. So, in the next lecture of this module we actually want to go beyond these graphical representations. Maybe they are pretty pictures but they're not always all that useful for us humans with our bounded minds to really grasp the uncertainty. And that will be the next step. So, let me wrap up this first lecture of the new module. We learned about the concept of the discreet random variable. We then introduced the concept of the probability distribution which really is nothing new it's just a little bit of new terminology giving probabilities to the numbers of a random variable and we discussed the representation either in tables and graphs. Thanks for being with us in this lecture and please come back for the next lecture. Thank you.
