Attaching a number to every outcome: a random variable as a function on the sample space

0:00Everything so far in this subject has measured sets. You take a collection of outcomes, you call it an event, and you ask how likely that collection is. Now watch something different in kind. Instead of drawing a ring around several outcomes, we are going to walk along the list of outcomes and write a number beside each one. One number per outcome. Nothing gets a ring. That is not an event. It is not a collection at all. It is a rule that takes an outcome and hands back a number, which is to say it is a function, and the set of outcomes is its domain.

0:42The name it is given is random variable, and both of those words are wrong. Nothing varies and nothing about the rule is random. We will come back to that at the end, because the name is the single biggest reason this idea is harder than it needs to be. For now, hold on to the picture. A ring around a subset, on the left. A number written against each outcome, on the right. Two different kinds of object, living on the same list.

1:16The definition is one sentence and it does three things at once. Split it open. First: the object is a function. Not a number, not a quantity that changes, not an event. A function. Second: what comes out is an ordinary number. Not a set, not a letter. A number you could write on the board. Third: what goes in is an outcome of the experiment. The domain is the sample space itself.

1:46Put those three together and you have the whole idea. A function from the outcomes of an experiment to the numbers. Everything else in this video is a consequence of those three lines, or a mistake people make about them. Of the three, exactly one cannot be relaxed, and it is hiding inside the word function. Each outcome gets exactly one number. Not at least one. Exactly one. Suppose someone hands you a rule that offers one outcome two different numbers, and says pick whichever you like. That is not a function, and it is not a random variable. There is no answer to give it; the right response is to refuse.

2:35Here is how much work that one word is doing. Take four outcomes and three allowed numbers, and count every rule that offers each outcome some set of those numbers. There are four thousand and ninety six such rules. Exactly eighty one of them are functions. Three thousand eight hundred and forty offer at least one outcome more than one number. A hundred and seventy five offer some outcome nothing at all.

3:05Read the definition loosely — anything that gives every outcome at least one number — and two thousand four hundred and one pass. That loose reading has just admitted two thousand three hundred and twenty rules that hand some outcome two numbers or more. So the word exactly is not decoration. It is doing almost all of the filtering. Take the smallest experiment that shows anything. A coin tossed twice, one toss after the other.

3:39Four outcomes, and the order matters: head then head, head then tail, tail then head, tail then tail. Now the first numbering. Count the heads. Head then head gets two. Head then tail gets one. Tail then head gets one. Tail then tail gets nought. Notice what was just done. Those four values were written out one outcome at a time, not described by a formula. That is what a function on a sample space is: a decision at every single outcome, settled before the coin is ever tossed.

4:17The numbers it produces are nought, one and two. Three numbers, from four outcomes. Three numbers from four outcomes. So two of the outcomes must have collided, and they did. Head then tail gets one. Tail then head also gets one. Two different outcomes, the same number. That is allowed. A function may send two inputs to the same output. What is forbidden is the reverse — one input going to two outputs — which is the demand we just spent a whole scene on.

4:54But allowed is not free. Here is the cost. If somebody tells you the number came out one, you cannot say which outcome happened. The number has thrown that away. Collect the outcomes that receive a given number and you get a grouping of the space: one outcome for nought, two outcomes for one, one outcome for two. The numbering merged a pair, and it is worth being able to say exactly which pair.

5:23The number and the outcome that produced it are different things. That sentence is the whole of this scene. Now keep the four outcomes exactly as they are and change nothing about the experiment. We are going to write a second number beside each one. This time: heads minus tails. Head then head gives two minus nought, which is two. Head then tail gives one minus one, which is nought. Tail then head, also nought. Tail then tail gives nought minus two, which is minus two.

6:00Two things just happened that the first numbering never did. One of the values is negative. Another is zero. So this cannot be counting anything. Counting does not go below nought. Whatever a numbering has to be, it does not have to count. And the more important point: the space did not change. The same four outcomes now carry a completely different set of numbers. There is no such thing as the right numbering for an experiment. There are as many as you care to write down.

6:38Two numberings on one space, and they disagree. Put them side by side and check: they agree at head then head, where both say two, and they disagree at the other three outcomes. But look harder, because this pair is not as different as it is being asked to seem. Take the first numbering, double it, and subtract two. Two doubles to four, minus two is two — correct. One doubles to two, minus two is nought — correct. One again, nought — correct. Nought doubles to nought, minus two is minus two — correct.

7:16Four outcomes, four hits. The second numbering is entirely rebuildable from the first, by one rule applied to its output. And it is not an accident of this space. Sweep every number of tosses from one up to eight — five hundred and ten outcomes in all — and heads minus tails is always twice the head count, less the number of tosses. Five hundred and ten hold, none fail. Break the shift and it fails at every single one of the five hundred and ten.

7:50So these two group the outcomes in exactly the same way. Each one can be rebuilt from the other. As a demonstration that a space does not fix its numbering, this pair is weaker than it looks. Let us repair that with a third numbering, and this one is mine. Give head then tail the number one. Give the other three outcomes nought. That is all of it. Now try to rebuild it from the head count. You cannot, and the reason takes one line.

8:24The head count gives head then tail the number one, and it gives tail then head the number one as well. Same number. But the new numbering gives them one and nought. Different numbers. Any rule applied to the head count's output sees a single one and would have to return two different answers. That is exactly the thing a function is not allowed to do. So this third numbering is genuinely independent of the first, in a way the other two are not. It separates two outcomes that the head count merges.

9:01And that is the sharp version of the lesson. The space does not determine the numbering, and here is a numbering that proves it without leaning on the first one at all. How unusual is that? Count them. Four outcomes, five allowed numbers, minus two up to two. Every way of hanging one of those five on each of the four: six hundred and twenty five numberings. Of those six hundred and twenty five, how many are rebuildable from the head count? A hundred and twenty five. Five hundred are not. So the first two numberings landed in the smaller group, and the third landed in the larger one.

9:47While we are counting, take the three things people believe a numbering must be. That different outcomes get different numbers: true of a hundred and twenty of the six hundred and twenty five, false of five hundred and five. The majority merge. That the numbers are never negative: five hundred and forty four of them go below nought somewhere. That it is really an event in disguise: only sixteen of the six hundred and twenty five take nothing but nought and one — and sixteen is exactly how many events a four outcome space has. Sixteen out of six hundred and twenty five. Events are a tiny corner of this.

10:32One more, because it is pretty. Six hundred and twenty five numberings, but only fifteen different ways of merging the outcomes. The numbering carries strictly more than the grouping does. Back to the name, because it has been waiting. Random. The rule is not random. It was fixed before anything happened, and it gives the same number every time the same outcome comes up. What is random is the outcome. The order matters: the outcome is random, and the rule is applied to whatever turns up.

11:11Variable. Nothing varies. The head count is not a quantity drifting about; it is a fixed table with four rows, written down in advance. And the silent third mistake, the one nobody warns about: this is not an event. An event is a set of outcomes, and this is a number against each outcome. Sixteen events on that space, six hundred and twenty five numberings. Different kinds of object. If the name were honest it would be something like outcome function. It is not going to change, so the thing to do is notice the name is a label and not a description.

11:55One last thing, and it is about where the ground ends. There is an obvious next question. You have the numbers, and you have a grouping of the outcomes behind each number. So how likely is each number? That question is not answered here, and this video is going to stop rather than pretend otherwise. Attaching a likelihood to each value, averaging those values, measuring how spread out they are — every one of those is a further step, and none of it follows from anything said above.

12:32What has been established is smaller and completely solid. A rule that gives every outcome exactly one number. Two outcomes may share a number, and if they do, the number no longer names the outcome. A space carries many numberings, six hundred and twenty five of them even on four outcomes with five values, and nothing picks one out as correct. Knowing precisely where an idea stops is part of knowing the idea. That is the last thing worth keeping.