Now it's time to look at some very specific probability distributions. And the ones that I've chosen here are probably the most frequently used probability distributions. And it is the case that there are hundreds of different probability distributions out there. And I've chosen three to present, but they are very very foundational. And if you were setting up a Monte Carlo simulation it's quite likely that a lot of the inputs would have one of the following distributions that I'm going to talk about now. So here's the most elemental of all the probability distributions, it's called the Bernoulli distribution. And what the Bernoulli distribution models for you is a random variable that can only take on one of two values. So, the easiest way to think about this is tossing a coin, it can come up as a head or a tail. But, more generally, we all represent the outcomes as either a one or a zero. So, you might equate a head to a one and a tail to a zero, and so we've got an event that can only take on one of two outcomes and we call the probability that x equal to one, we give that the letter p in general. And because something has to happen that means the probability x equal to zero is one minus p and so that's a very sort of elemental or a comic random variable from which we'd be able to build up lots of more Interesting and complicated probability distributions. So often times when we talk about these Bernoulli random variables we view them as an outcome of an experiment and that outcome can either be a success or a failure. And in a business setting that might be whether an individual buys your product or doesn't buy your product. And as an example of a Benoulli random variable. Now, once we got this probability distribution written down, we're able to calculate the summaries. So I'm going to introduce the summaries now. The first one is mu, the mean. Another way of writing that is the expected value of X, the average or the expected value. So you can see E of X is another way of denoting the mean, the expected value of X and the way that you get the expected value of one of these discrete random variables Is by saying, what values can it take, and what are the probability that it takes on those specific value and then add them up. And so this Bernoulli can take on the value one with probability p, or it can take on the value zero with probability one minus p. If you do that algebra, you'll find that the mean of this Bernoulli random variable is p. Now the definition of the variance sigma squared is the expected value of X minus mu squared. So you can see through this formula why the variance is able to pick up the spread. Because if you look inside the parenthesis, you'll see mu there, that's the center of the distribution. And X- mu is telling you how far a particular value of the random variable is away from the center. So how far are you from the center and the expected value of that quantity squared is sort of how far on average are you from the center, but on a squared level. So using a square distance. Now if you work that quantity out the expectation calculation is the same idea. You figure out the values that X- mu can take and then you multiply those values by the probabilities and so if we work that out, we'll see that X- mu, that would be 1- p, X- mu squared times p is the first term because if X = 1, then you just plug 1 in and you get 1- mu squared, and what's the probability that X = 1? That's p, so that's the first term, 1- p squared times p, and then the other possible outcome is for X to equal 0. And if you put 0 in and you do 0-mu, but remember mu, the mean is just p from the original, the previous calculation. You get 0-p all squared, times the probability that you got a 0, which is 1-p. And so you can see the two probabilities there, p and 1- p. And you can see the two components. What's X- mu squared? It's either 1 p all squared or 0- p all squared. You take that. You do your algebra, and you end up with p times 1- p. So that is the variance of a Bernoulli. And if we want to get the standard deviation of this particular random variable, then that's easy. You just take the square root of the variance. And so those are the calculations associated with finding a mean and a variance or standard deviation. If you're wondering why we have these two different measures of spread, the variance and the standard deviation, it's really about the units associated with them. So let's say X was measured in yen, for the sake of argument, then the variance would be measured in yen squared, so it's a little bit hard to get your head around the idea of a squared yen. But if you square root that and you look at the units of measurement, if the variance is measured in yen squared, then the standard deviation would be measured in yen, so it's back on the scale of the data, so it's easier to interpret the standard deviation. So, those are our calculations for the key summaries, and just to do a particular example, what if we have p = 0.5? So that's a very special instance of the Bernoulli, where each outcome is equally likely. That's tossing a fair coin. Then you'll find that for p = 0.5, the mean is just equal to 0.5 itself, remember the mean is equal to p. The variance is p times 1-p, a half times a half is a quarter, and then the standard deviation if you square root the quarter, you are going to get a half, and so, those are the key summaries. And, this process of creating the summaries is very useful as I say at the end of having done something like a Monte Carlo simulation to say, here's why I think the answer is going to be mu and then we can use the standard deviation to give us a sense of the variability about the mean. So here's a example of where this Bernoulli distribution could come in useful. And I'm going back to one of the initial examples of this class, where we're talking about drug development and creating a probabilistic model would potentially involve these Bernoulli random variables. So we might come up with a scenario where we think the outcome of whether or not a drug is approved by some regulatory body is a Bernoulli random variable. It's either approved yes or it's not approved, no. We can write those as the values 1 and 0 so we're back to this Bernoulli setup. And then our experts might have estimated that the probability that we're a yes, that we get approved, is 0.65 and the probability we don't get approved is 0.35. Notice those two add up to one. Now if we believe that given the drug is approved, the projected revenue is $500 million from it, zero otherwise because if it's not approved we can't sell it, so we'll get no revenue at all. We could work out the expected revenue. And the calculation of these expectations is as before you take the different realizations of the random variable and the revenues under each realization. More applied through the probabilities of those events. So if we get approved, which is probability 0.65, we get 500 million revenue. And if we don't get approved we get zero revenue, and that happens with probability 0.35. We add those up and we get an expected revenue of $325 million. And so this Bernoulli could potentially be used as a building block for a Monte Carlo simulation to get a sense of, if we've got ten of these drugs what's the expected revenue going to be?