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

What to assume they know

• Sample space and outcome, and an event written as a set of outcomes
• Function, domain and codomain, and the requirement that a function assign exactly one output to each input, from Chapter 1
• The difference between a function that is one-one and one that merges inputs, also from Chapter 1
• The two-toss coin experiment, which the chapter has used repeatedly by now
• Negative integers, and reading a difference of two counts

What they should be able to do

• State what kind of object a random variable is, in terms of domain and outputs
• Say why the object is a function and not a number, an event or a quantity that changes
• Check that a proposed assignment really is a function on the given space
• Write down the value of a given assignment at each outcome of a small space
• Recognise that two different outcomes may be given the same number, and say what that costs
• Recognise that the numbers may be negative or zero
• Construct a second assignment on the same space and show that the two disagree somewhere
• State precisely where the chapter's treatment ends, and name the material its own introduction promises but does not contain

Where it usually goes wrong

• "A random variable is a variable." Nothing varies. It is a fixed rule that was decided before the experiment ran, and it produces the same number every time the same outcome comes up.
• "A random variable is random." The rule is not random. The outcome is random, and the rule is applied to whatever outcome turns up. Say it in that order; reversing it is where the confusion lives.
• "A random variable is an event." An event is a set of outcomes; this is a number attached to each outcome. They are different kinds of object, and the chapter has spent twenty-five pages on the first kind before introducing the second.
• "It has to count something." It has to assign a number. The chapter's second example takes a difference and goes negative, which counting cannot do.
• "Different outcomes must get different numbers." Two of the four outcomes in the chapter's own example get the same number. A function is allowed to merge inputs; only the reverse is forbidden.
• "One outcome could get two numbers if the rule is ambiguous." Then it is not a function and not a random variable. This is the one demand that cannot be relaxed, and it is worth checking out loud against the four printed values.
• "There is one right random variable for an experiment." There are as many as you care to define. The whole reason the chapter prints a second one is to refuse this.
• "The next step is obviously to work out the chance of each value." It is the obvious next step and this chapter does not take it. An explanation that takes it has left the book behind; see Notes for why that matters here more than usual.

Questions to check understanding

• State what a random variable is, naming its domain and the kind of thing its outputs are
• Given a small sample space and a described rule, write out the value at every outcome
• Decide whether a proposed rule is a random variable, with at least one distractor that fails to be a function
• Given the chapter's two-toss space, define a second assignment and name an outcome at which it disagrees with the first
• Say what it means when two outcomes receive the same number, and whether that is allowed
• Explain in one sentence why the name is misleading, addressing both words
• State what this chapter does and does not go on to do with a random variable

Examples worth working on the board

Values marked verified are worked out here from the chapter's own printed data; neither answers file was opened, and this chapter prints no answers to its exercises.

• Where the material sits (Part II pp. 430–431). Read off the printed pages: it is a block headed Remark, set in the same style as the chapter's three other remarks, beginning at the foot of Part II p. 430 immediately below the last reversal example and running over onto the top of Part II p. 431, where it ends, after a gap, at the start of Exercise 13.3. Thirteen lines in all, counted off the two printed pages. It carries no section number and no subsection number. Anyone establishing this chapter's structure by scanning for numbered headings will not find it, which is exactly how it comes to be reported as deleted when it is not. See Notes.
• The defining sentence (Part II p. 430). One sentence, and it does three things at once: it says the object is a function; it says the outputs are ordinary numbers; and it says the inputs are the outcomes of the experiment. Split it into three and tick each part against the example that follows.
• The experiment (Part II pp. 430–431). A coin tossed twice, one toss after the other, giving four outcomes. The chapter has used this space several times already, which is the right choice — nothing about the space is new, only what is being done to it.
• The first assignment (Part II p. 431). Count the heads. Verified against the four printed values: two heads gives two; a head then a tail gives one; a tail then a head gives one; two tails gives nought. All four are written out individually rather than described by a rule, which is worth mirroring — the point is that a function is settled outcome by outcome.
• Two outcomes sharing a value. An added observation, not the chapter's. Two of the four outcomes are both sent to one. So this assignment merges outcomes, and knowing its value does not in general tell you which outcome occurred. That is allowed — a function may do this — and it is worth naming, because it is the difference between the number and the outcome that produced it.
• The second assignment (Part II p. 431). Heads minus tails. Verified against the four printed values: two heads gives two; each mixed outcome gives nought; two tails gives minus two. This is the chapter's demonstration that the space does not determine the numbering. It also, quietly, makes two further points the chapter does not draw out: the outputs may be negative, and they may be zero.
• How different the two really are. Working added here, and worth ninety seconds. On this particular space the second assignment is completely determined by the first — it is twice the first, less two, at every one of the four outcomes. Verified at all four: two gives two, one gives nought, one gives nought, nought gives minus two. So the chapter's pair are different functions but not independent ideas, and a sharp student will notice. Supply a third assignment that genuinely is not recoverable from the first: send the outcome head-then-tail to one and the other three to nought. Verified: the first assignment gives the same number, one, to head-then-tail and to tail-then-head, while this third one separates them, so no rule applied to the first assignment's output can reproduce it. Label it clearly as not printed in the book.
• The closing sentence (Part II p. 431). One line saying the two are different assignments living on the same space. That is the last thing the Remark says, and Exercise 13.3 begins immediately afterwards.
• What is not there. Checked on the page image of every one of the thirty-three pages and by searching the extracted text of all of them: no exercise item anywhere in the chapter uses a random variable, the Summary on Part II p. 437 carries no bullet about one, and the words for the average and the spread of a distribution each occur exactly once in the whole chapter, on the introduction page. See Notes.

Figures to have open

• A four-row outcome table, created empty in section 4 and filled across sections 5, 7 and 8. It must be one table gaining columns, not three tables — the entire argument is that the space stayed fixed while the numbering changed. Build with the repo's DataTable component and give all four rows one shared fitSize.
• A single side-by-side panel for section 2: the same four outcomes drawn twice, once with a ring around a subset and once with a number beside each. This is an added device and it carries the whole distinction; the chapter draws nothing at all for this Remark.
• No figure for section 3. The definition is text and Statement carries it.
• Section 10 needs no illustration and must not have one. It is a boundary statement, and decorating it invites the student to expect a continuation.

Where this sits in the book

• NCERT Class 12 Mathematics, Chapter 13 "Probability", the unnumbered Remark defining a random variable, Part II p. 430
• The two-toss experiment, the first assignment's four values, the second assignment and its four values, and the closing sentence, Part II p. 431
• §13.1 Introduction, where the term is announced along with three things the chapter does not contain, Part II p. 406
• Exercise 13.3, which begins directly beneath the Remark and never refers to it, Part II pp. 431–433
• Summary, Part II p. 437, which omits the Remark entirely
• Historical Note, Part II p. 438, the only other place a named distribution appears