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IB DP Maths AI · Topic 4 Statistics and probability · AHL 4.14–4.19 · ~8 min read

Estimation, Poisson, hypothesis tests and Markov chains

HL statistics for Maths AI: linear combinations of random variables, unbiased estimates, the central limit theorem, confidence intervals for a mean, the Poisson distribution, hypothesis tests with binomial, Poisson and normal distributions, Type I and Type II errors, and Markov chains with transition matrices and steady states.

E(aX + b) & Var(aX + b)Linear combinationsUnbiased estimatorsCentral limit theoremConfidence intervals (z and t)Poisson distributionBinomial hypothesis testsCritical regions
RSby Riyaz Sir
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No tag = SL and HLHL ONLY HL students onlyFORMULA BOOKLET given in the examM1A1 how marks are awardedhighlight = comes up again and againTest yourself at the end

In 30 seconds

  • Combining variables: E(aX + bY) = aE(X) + bE(Y); for independent X and Y, Var(aX + bY) = a2Var(X) + b2Var(Y). Variances add even when you subtract.
  • Central limit theorem: for large n (n ≥ 30 is the usual guide), the sample mean X̄ is approximately N(μ, σ2n), whatever the shape of the population.
  • Confidence interval for μ: x̄ ± zσ√n when σ is known, or x̄ ± tsn−1√n when it isn't.
  • Poisson Po(m): counts of random events in a fixed interval, with mean = variance = m.
  • Errors: a Type I error rejects a true H0 (its probability is the significance level); a Type II error fails to reject a false H0. A Markov chain moves between states with a transition matrix T; the steady state s satisfies Ts = s.

By the end you can

  • Use E(aX + b), Var(aX + b) and linear combinations of independent random variables. HL
  • Find unbiased estimates of the mean and variance. HL
  • Use the central limit theorem and find confidence intervals for a mean. HL
  • Use the Poisson distribution, including sums of independent Poisson variables. HL
  • Carry out hypothesis tests with binomial, Poisson and normal distributions, find critical regions, and Type I and II error probabilities. HL
  • Use transition matrices for Markov chains and find steady states. HL
Syllabus noteFrom first exams in 2029, the Poisson distribution and the hypothesis tests for a normal mean, a Poisson mean and the correlation coefficient leave AI HL. If you sit exams up to November 2028, you still need all of this note.

1Combining random variables HL

E(aX + b) = aE(X) + b, Var(aX + b) = a2Var(X) FORMULA BOOKLET
adding b doesn't change the spread
E(a1X1 ± a2X2) = a1E(X1) ± a2E(X2), Var(a1X1 ± a2X2) = a12Var(X1) + a22Var(X2) FORMULA BOOKLET
X1, X2 independent
Worked example

X ~ N(50, 32) and Y ~ N(30, 42) are independent. Find the distribution of X − Y and P(X − Y > 25).

  1. E(X − Y) = 20, Var(X − Y) = 9 + 16 = 25 M1A1
  2. X − Y ~ N(20, 52) A1
  3. P(X − Y > 25) = 0.159 A1
Common mistakeWriting Var(X − Y) = 9 − 16. Variances never subtract: two sources of variation always add uncertainty.
  • Unbiased estimates: x̄ estimates μ; sn−12 = nn − 1sn2 estimates σ2. For 10, 12, 9, 11, 13: x̄ = 11 and sn−12 = 104 = 2.5.

2The central limit theorem and confidence intervals HL

405060708090100population: σ = 12X̄ for n = 36: σ/√n = 2Sample means cluster much more tightly around μ, and are approximately normal for large n.
The distribution of the sample mean is centred at μ and much narrower than the population.
Worked example

A population has mean 70 and SD 12. A sample of 36 is taken. Find P(X̄ > 73).

  1. X̄ ~ N(70, 12236) = N(70, 22) (CLT) M1A1
  2. P(X̄ > 73) = 0.0668 A1
x̄ ± z × σ√n, x̄ ± t × sn−1√n FORMULA BOOKLET
confidence intervals for μ
true μ (unknown)44464850528 samples, 8 intervals: 7 contain μ (blue) and 1 misses it (red).
A 95% interval method captures the true mean in about 95% of samples. Any single interval either contains μ or it doesn't.
Worked example

A sample of 25 has x̄ = 48.2 and sn−1 = 5. Find a 95% confidence interval for μ.

  1. σ unknown, so use t with 24 df: t = 2.064 M1
  2. 48.2 ± 2.064 × 55 M1
  3. 46.1 < μ < 50.3 A1
  • Interpreting: "we are 95% confident μ is between 46.1 and 50.3". A wider interval comes from a higher confidence level or a smaller sample.

3The Poisson distribution HL

012345678910X ~ Po(4): calls per hourmean = variance = 4red: P(X ≥ 6) = 0.215
Po(4): the number of calls in an hour. The mean is 4 and so is the variance.
P(X = x) = mxe−mx!, E(X) = Var(X) = m FORMULA BOOKLET
Poisson with mean m
  • Conditions: events happen singly, independently, at a constant average rate. If X ~ Po(m) and Y ~ Po(n) are independent, X + Y ~ Po(m + n).
Worked example

A helpline gets 4 calls an hour on average. Find P(exactly 2 calls in an hour), P(at least 6 calls in an hour), and P(no calls in 30 minutes).

  1. X ~ Po(4): P(X = 2) = 0.147 A1
  2. P(X ≥ 6) = 1 − P(X ≤ 5) = 0.215 A1
  3. in 30 minutes Y ~ Po(2): P(Y = 0) = e−2 = 0.135 M1A1
Exam tipA quick check for Poisson data: the sample mean and variance should be roughly equal.

4Hypothesis tests HL

Worked example

A seed company says 30% of its seeds fail. A gardener suspects more fail. In a sample of 20, 10 fail. Test at the 5% level.

  1. X ~ B(20, p); H0: p = 0.3, H1: p > 0.3 A1
  2. P(X ≥ 10 ∣ p = 0.3) = 1 − P(X ≤ 9) = 0.0480 M1A1
  3. 0.0480 < 0.05, so reject H0: there is evidence that more than 30% fail R1
  • Critical region: the set of results that lead to rejecting H0. Here P(X ≥ 10) = 0.048 ≤ 0.05 but P(X ≥ 9) = 0.113 > 0.05, so the critical region is X ≥ 10.
02468101214161820critical region X ≥ 10blue: p = 0.3 (H₀)red: Type I error 0.048orange: p = 0.5dark orange: Type II 0.412Type I: reject a true H₀. Type II: fail to reject a false H₀.
The Type I error is the red tail under H₀; the Type II error is the orange part of the true distribution outside the critical region.
Worked example

For this test, find the probability of a Type I error, and of a Type II error if really p = 0.5.

  1. Type I: P(X ≥ 10 ∣ p = 0.3) = 0.0480 A1
  2. Type II: P(X ≤ 9 ∣ p = 0.5) = 0.412 M1A1
  • Other tests: a normal mean test uses X̄ ~ N(μ0, σ2/n) (z-test) or a t-test when σ is unknown; a Poisson mean test works like the binomial one; a correlation test checks H0: ρ = 0 using r and the GDC's linear regression t-test. All three are removed for 2029.
Common mistakeUsing a discrete distribution as if it hits exactly 5%. With binomial and Poisson tests, the actual significance level is the probability of the critical region (here 4.80%), usually below 5%.

5Markov chains HL

SunnyRainy0.20.40.80.6T = (0.8 0.4; 0.2 0.6): column = from, row = to. Steady state: ⅔ sunny, ⅓ rainy.
A two-state Markov chain. Each column of T adds to 1.
sn = Tns0
state after n steps; the steady state s satisfies Ts = s
Worked example

Sunny days are followed by sunny days with probability 0.8, rainy days by sunny days with probability 0.4. Today is sunny. Find the probability that it is sunny in 2 days' time, and the long-term proportion of sunny days.

  1. T = (0.8 0.4; 0.2 0.6), s0 = (1; 0) A1
  2. T2s0 = (0.72; 0.28), so P(sunny) = 0.72 M1A1
  3. steady state: 0.8s + 0.4(1 − s) = s ⇒ s = 23, so ⅔ of days are sunny in the long run M1A1
  • Eigenvector link: the steady state is the eigenvector of T with eigenvalue 1, scaled so its entries add to 1.
Linking questionLinear combinations generalise the binomial and normal ideas of Probability, binomial and normal distributions; the SL χ² and t-tests (Chi-squared and t-tests) use the same logic as these HL tests; Markov chains use eigenvectors from Matrices and eigenvalues and random walks from Graph theory.
Idea for your IA / explorationEstimate a real population mean (the battery life of phones in your class, the waiting time at a canteen) with a confidence interval, or model customer loyalty between brands as a Markov chain and find its steady state.

Test yourself — cover the answers first

Quick check

  1. HL Var(X) = 4. Find Var(3X − 2).
  2. HL X and Y are independent with variances 5 and 7. Find Var(X − Y).
  3. HL What does the CLT say about X̄ for large n?
  4. HL X ~ Po(3). Find Var(X).
  5. HL What is a Type I error?
  6. HL What must each column of a transition matrix add up to?

Exam-style

  1. HL Cars pass a point at an average of 2 per minute and lorries at 0.5 per minute, independently. Find the probability that exactly 3 vehicles pass in one minute.
  2. HL Find a 90% confidence interval for μ from a sample of 50 with x̄ = 120 when σ = 15 is known.
Show answers
  1. 36
  2. 12
  3. approximately N(μ, σ²/n)
  4. 3
  5. rejecting H₀ when it is true
  6. 1
  7. total V ~ Po(2.5)
    P(V = 3) = 2.53e−2.53!
    = 0.214
  8. z = 1.645
    120 ± 1.645 × 15√50 = 120 ± 3.49
    116.5 < μ < 123.5
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Independent notes written by Riyaz Sir. Not produced, endorsed or approved by the International Baccalaureate Organization. Syllabus codes refer to the IB DP subject guides.