maths

Probability & Statistics: Bayes, Distributions

Conditional probability, Bayes' theorem, and the Binomial/Poisson/Normal trio — the highest-frequency quant questions for all branches.

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Key formulas

  • P(A|B) = P(B|A)·P(A) / [P(B|A)P(A) + P(B|A′)P(A′)]

    Bayes' theorem: The classifier formula — prior × likelihood normalised by total probability.

  • P(X=k) = C(n,k)p^k(1−p)^(n−k); Poisson: P(X=k) = λ^k·e^(−λ)/k!

    Binomial / Poisson: Discrete counting distributions; Poisson with mean = variance = λ.

  • Z = (X − μ)/σ ~ N(0,1)

    Standardisation: Convert any normal variable to the standard normal for probability tables.

Cheat sheet

  • ▸Conditional probability P(A|B) = P(A∩B)/P(B). Bayes: P(A|B) = P(B|A)·P(A)/P(B) — inverts 'which cause explains the evidence'; the denominator expands via total probability.
  • ▸Independence: P(A∩B) = P(A)·P(B). Never confuse mutually exclusive (P(A∩B) = 0, can't both happen) with independent (both happening tells nothing about the other).
  • ▸Binomial: n trials, success p → P(X = k) = C(n,k)p^k(1−p)^(n−k); mean np, variance np(1−p).
  • ▸Poisson: rare events per interval, λ = mean = variance; the limiting case of binomial when n large, p small (λ = np).
  • ▸Normal: bell curve, mean μ, std σ; 68–95–99.7 within 1σ, 2σ, 3σ. Standardise Z = (X − μ)/σ to use tables; Binomial/Poisson → Normal for large n (np, n(1−p) > 5).
  • ▸Expected value is linear: E[aX + bY] = aE[X] + bE[Y] always. Variance adds only for independent variables.

When to use which distribution

SituationDistributionMean / Variance
n independent yes/no trialsBinomialnp / np(1−p)
Rare events in time/spacePoissonλ / λ
Sum of many small effectsNormalμ / σ²
Waiting time between Poisson eventsExponential1/λ / 1/λ²

Visual

Normal curve and the empirical rule:
        ___
      /      \      68% within ±1σ
     /|      |\     95% within ±2σ
    / |      | \    99.7% within ±3σ
   3σ 2σ  μ 1σ ...  (Z = (X−μ)/σ)

Top 4 campus interview questions

MCQ mastery quiz

1. In Bayes' theorem, the denominator represents…

2. For a Poisson distribution, the relation of mean and variance is…

3. About 95% of a normal distribution lies within…

4. Events A and B are independent when…