A random variable is a rule that turns an outcome into a number. Everything else in this unit is arithmetic done on those numbers.
Which values can it take?
A coin is tossed three times. X is the number of heads.
Three tosses can give no heads at all, so 0, 1, 2, 3. A forgets that TTT is possible — the commonest slip, and it costs a whole outcome. B lists the outcomes, not the variable: a random variable is always a number. D would be right for a continuous variable, but you cannot toss 1.7 heads.
The experiment produces outcomes — heads, tails, a colour, a person. The random variable is a rule that attaches a number to each one. Write the rule as a sentence starting “X is the number of…” and the rest of the topic follows.
Notice that several outcomes can share a value. That is exactly why probabilities are not all equal, and it is the whole content of the next lesson.
The test is not whether the values are whole numbers — a peso amount like ₱12.50 is still counted, in fifty-centavo steps. The test is whether there is a gap between one possible value and the next. Between 41 kg and 42 kg there is no gap; a sack can weigh 41.3874 kg.
A shoe size of 8½ is discrete: there is nothing between 8½ and 9. And a count can be huge and still discrete — the population of a barangay has no value between 4 112 and 4 113. Ask whether a value between two neighbours is possible, not whether the numbers look tidy.
smallest total: 1 + 1 = 2
not 0 — a die has no zero
largest total: 6 + 6 = 12
both at their maximum
X = 2, 3, 4, …, 12
eleven values, not twelve
Find the extreme cases first and fill in between. Starting a list at 1 out of habit is the usual error here, and it survives all the way to a wrong probability table.
| Written | Means |
|---|---|
| X | the random variable itself, as a rule |
| x | one particular value it took |
| P(X = 2) | the probability the variable equals 2 |
| P(X ≥ 2) | the probability it is 2 or more |
Capital X is the rule; small x is a number it produced. Examiners lean on that difference, and so does every formula from here on — ∑x·P(x) makes no sense until you can see which symbol is which.
fewest red: both blue → X = 0
possible, there are 2 blue
most red: both red → X = 2
possible, there are 3 red
X = 0, 1, 2
two draws, so 3 is impossible
Had the box held only 1 red, the list would stop at 1. Always check the extremes against what the box actually contains rather than against the number of draws.
The same routine — smallest, largest, fill in — on rules that behave less tidily.
Two dice are rolled. X is the larger number minus the smaller. List the values.
smallest: equal faces, e.g. 4 and 4 → X = 0
a difference can be zero
largest: 6 and 1 → X = 5
the faces furthest apart
X = 0, 1, 2, 3, 4, 5
six values
Same dice as the totals, a different rule, so a different list.
A coin is tossed until the first head. X is the number of tosses.
smallest: head on the first toss → X = 1
at least one toss is made
largest: none — tails can run on
TTTT…H
X = 1, 2, 3, … (discrete)
still counted, with gaps between values
Endless values do not make a variable continuous; gaps between neighbours keep it discrete.
A game: two coin tosses; win ₱10 per head, lose ₱5 per tail. X is the net winnings.
HH: 10 + 10 = 20
best case
HT or TH: 10 − 5 = 5
two outcomes, one value
TT: −5 − 5 = −10
a loss is a negative value
X = −10, 5, 20
three values from four outcomes
Money won or lost is a signed number; write a loss as negative, not as a second positive list.
Expect two kinds of item: classify a variable as discrete or continuous, and list the values a stated X can take. Test classification by asking whether a value between two neighbours is possible; list values by finding the smallest and largest case first, then filling in. The 10-second check: test both ends against the situation — can 0 happen, and is the top value really reachable?
Discrete or continuous?
A sari-sari store records the weight of rice in each sack it sells.
Weight is measured, and between 24.9 kg and 25.0 kg there are infinitely many possible weights, so it is continuous. A confuses recording with measuring — rounding to one decimal is what the scale prints, not what the sack weighs. C counts sacks, which is a different variable and would indeed be discrete.
Hint: ask whether a value strictly between two neighbouring ones is possible. If it is, the variable is continuous, however the number happens to be written down.
