Indicator random variables
An indicator 1_A takes the value 1 when A occurs and 0 otherwise, so E[1_A] = P(A) — expectation and probability become the same computation. Since 1_A² = 1_A, the variance is P(A)(1 − P(A)). Writing any count as a sum of indicators and applying linearity is the single most productive technique in discrete probability.
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Method: how to approach it
The order below is what examiners expect to see, and each step carries its own marks.
- Define one indicator per eventXᵢ = 1 if the ith thing happens. Be precise about what "the ith thing" is.
- Express the count as their sumX = ΣXᵢ. Verify that the sum really counts what you want.
- Use E[Xᵢ] = P(event i)The expectation of an indicator is just a probability, so no distribution work is needed.
- Add for the expectation; add covariances for the varianceVar(ΣXᵢ) = ΣVar(Xᵢ) + 2ΣCov(Xᵢ,Xⱼ), and the covariances usually do not vanish.
Worked example
A fair die is rolled 60 times. Find the expected number of sixes, using indicators.
- Let Xᵢ = 1 if roll i is a six, for i = 1 … 60.
- The total number of sixes is X = X₁ + … + X₆₀.
- E[Xᵢ] = P(six on roll i) = 1/6.
- By linearity, E[X] = 60 × 1/6.
Answer. E[X] = 10 — agreeing with the binomial mean np = 60 × 1/6.
Where marks get dropped
These are the specific errors that cost credit on indicator random variables questions — QED's rubric penalises each of them separately.
- Defining indicators for events that overlap in a way the sum does not intend. Check that ΣXᵢ counts each occurrence exactly once.
- Computing E[1_A] as something other than P(A). The indicator takes only the values 0 and 1, so its mean IS the probability.
- Adding variances of indicators without covariances when the events are dependent.
Practise this until it is automatic
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Indicator random variables — frequently asked questions
Why is E[1_A] = P(A)?
Because E[1_A] = 1·P(A) + 0·P(Aᶜ) = P(A). The definition of expectation collapses to the probability.
What is the variance of an indicator?
p(1 − p) where p = P(A), since E[1_A²] = E[1_A] = p and Var = p − p².
Where does this technique shine?
Expected number of fixed points in a permutation, expected collisions in hashing, expected inversions in a random list — all become one line.
The rest of Probability
Events, conditional probability, random variables. Each subtopic below has its own method, worked example and mark-losing traps.
- 1Sample spaces & events
- 2Conditional probability & independence
- 3Bayes’ theorem
- 4Random variables & expected value
- 5Variance & standard deviation
- 6Binomial & uniform distributions
- 7Law of total probability & tree diagrams
- 8Geometric & Poisson distributions
- 9Joint, marginal & conditional distributions
- 10Markov & Chebyshev bounds
- 11Equally likely outcomes & counting
- 12Linearity of expectation
- 13Indicator random variables
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