So, far this week, we've seen a few examples of some very simple probability distributions ranging from things like, tossing a fair coin, or to rolling a fair die. What I like to consider now, is our first example of a family of probability distributions. So, remember up to this point, we've generally considered examples whereby the outcomes themselves, the different members of the sample space recall, happened to be equally likely. But of course, there'll be many situations in life when that would not be the case. So, to illustrate this point and to develop our first family of probability distributions let's return to the toss of a fair coin. So, we said that there were two possible outcomes, heads and tails. Well, let's now extend this to situations where we're not just necessarily looking at tossing a coin, but situations where there is a binary set of outcomes. What we may wish to call pap success and failure. Now, many examples we could give where the world has this sort of dichotomous nature. A dichotomy is when you just have those two possible situations. So, for example it might be you sit an exam, you either pass it or you fail it. We said about tossing a coin of heads or tails. You go for a job interview, get the job, you don't get the job. Maybe we can consider medical statistics, looking at survival rates. You have an operation, you survive or you don't survive. There are no other possible outcomes in those particular situations. So we may consider success and failure, and let's consider again this binary coding of one and zero. So let some random variable X take the value one when a success occurs, and zero when a failure occurs. Now, these labels of success and failure don't necessarily have to correspond to good and bad things necessarily. If we are considering tossing a coin, which is sort of the better outcome, heads or tails, it's a notional success and failure. But nonetheless, we have this partition, this separation into a success and a failure. Now, if it was a fair coin consider thus far, then those two outcomes would deem to be equally likely. We attach probabilities of a half and a half to the success and failure respectively. But, I'd like to extend this discussion now because clearly if you sit an exam, you're not necessarily equally likely to pass it or to fail it. Depending on how prepared you are, if you studied a lot throughout the year, of course the probability to pass it is no doubt going to be much greater than for failing it. Similarly, if you do no work all year and you just turn up to the exam hoping to wing it, the chances of failure of course are going to be much higher than passing the exam. So, let's now introduce the concept of a parameter. So for this, I'm going to use the Greek letter pi. Now, please don't be confused with the use of pi, you may have seen it in high school math when you're calculating the area of a circle, here we're not considering that mathematical content of 3.14 et cetera, et cetera, rather here pi represents the probability of success. So, we have this random variable X taking the values of zero for failure, one for success, and pi is going to be the probability that X is equal to one. Now we said in a sample space, this was the set of all possible outcomes in a random experiment. Indeed we can think of this as all possible values of these random variable X. Here only zero or one. Similarly, a parameter can also have a space namely the parameter space, the possible values it can take. So given we previously defined probability to be some value over the unit interval from zero to one, then this represents the parameter space of pi. So, if we have pi as the probability of success, and given there are only two possible outcomes which themselves are mutually exclusive, you cannot have a success and a failure at the same point in time. They are also collectively exhaustive because in this two point distribution, you either have a success or a failure. There is no other possible outcome. And if pi is the probability of success, it follows that one minus pi must be the probability of failure. So, we now see our first form of family of probability distributions. In the next section, we're going to see a selection of different probability distributions, but this is my personal favorite. This is the Bernoulli the distribution. Now, the Bernoulli is where a very famous family of Swiss mathematicians, scientists. The Bernoulli distribution is named after Jacob Bernoulli. So, I love this because there are many things in life that we can easily reduce to situations where there are simply two possible outcomes. For example, even in an election where there are perhaps several political candidates to choose from, you could easily reduce this to a two-point situation, where you either vote for one specific candidate or one of the others. So the Bernoulli distribution has many possible applications, more of which we will see later on in the course. So, this family of probability distributions, the Bernoulli family, suggesting there are different members of the Bernoulli family. Presumably, there's like a mommy, a daddy, grandparents, children as well. Well, we've already seen one member of this Bernoulli family, and that was when we were tossing that fair coin, because there we said those two outcomes, the success and failure normally let's say heads and tails were deemed to be equally likely and hence pi, the probability of success would be equal to 0.5. And by extension, the probability of failure one minus pi would also be 0.5. So, this represents one member of that Bernoulli family. But of course as a family of distributions, there were different values that this pi parameter can take. The only constraint is that it must lie between zero and one given it reflects a probability. So, let's imagine we were looking at the human population. Now, let's consider blood type. That varies from a person to person. So there we have particular proportion of the population with a specific blood type. Now, there are many different blood types out there, but we could consider one specific blood type versus all other blood types. So, if we were interested in the prevalence of a particular blood type in the population as a whole, maybe for blood transfusion purposes say, then we might call success having the blood type of a specific type. And hence pi could represent, although it's a probability, we could view it now as a proportion namely, the proportion of the whole population which has that particular blood type. Now of course, we can consider situations where that probability of success differs quite a lot from 0.5. And indeed, this is giving rise to our first family of probability distributions, for which each different member of that family is distinguished by a different value for that particular parameter. Now, perhaps we'll just conclude this session with a revisit of those expectations we saw a little earlier. So, in the simple examples considered so far, we said an expected value of some random variable X represented a probability weighted average. Whereby, in the examples so far, we've taken each value of the variable and multiplied by its corresponding probability of occurrence, and added those things together. So, let's consider the expected outcome for a generic Bernoulli distribution. So, we have those two outcomes, zero and one with probabilities of 1−π and π respectively. So, 0×(1−π)+1×π will give you π at either probability of success. So, the expected value of a Bernoulli distribution is simply the prevalence of the success type within that population under consideration. So, one can consider this taking the long term approach to interpreting expectations. Remember the idea of a long run at average that the pi, the probability of success, represents the proportion of successes which would occur. So, if we for example took a random sample from this distribution, and more on random sampling later on in the course, some of the times we would get successes, some of the times we would get failures. Of course the probability of success is pi, the probability of failure is 1−π, and in the long run we would anticipate to get as a proportion of success is that value π. So, this concept of a parameter is very important. We're just sort of touching the tip of the iceberg here with respect to parameters. Much more on that going forward later on in the course.