Lesson 5 of 14 · All lessons in this module
  1. Random variables: PMF, PDF and CDF
  2. Expected value, variance and standard deviationVideo lesson
  3. Variance and covariance shortcuts, and the law of large numbers
  4. Discrete distributions: uniform, binomial and Poisson
  5. Continuous uniform and normal distributions, and simulating random draws
  6. Log-normal and logistic distributions, and the moments of key distributions
  7. Integrated cases: reading probabilities and moments from a return model
  8. Conditional probability and probability trees
  9. Conditional expected values and variancesVideo lesson
  10. Covariance from a joint probability function
  11. Joint, marginal and conditional distributions as information arrives
  12. Bayes' formula: updating a probability with new informationVideo lesson
  13. Bayes in practice: diffuse priors, missing inputs and screening tests
  14. Integrated cases: a tree with model forecasts and a Bayes update

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Lesson 5 of 14 · 14 min

Continuous uniform and normal distributions, and simulating random draws

The continuous uniform spreads probability evenly over an interval and is the raw material of simulation; the normal is the symmetric bell that finance uses as its default model of returns.

In short

  • Continuous uniform on [a, b]: flat density 1/(b−a)1/(b-a), mean (a+b)/2(a+b)/2, variance (b−a)2/12(b-a)^2/12, skewness 0, kurtosis 1.8.
  • Under a uniform, probability is proportional to interval length: P(x1≤X≤x2)=(x2−x1)/(b−a)P(x_1 \le X \le x_2) = (x_2 - x_1)/(b - a).
  • Inverse-CDF method: draw u from U(0, 1), then take the value x whose CDF equals u. Uniform draws become draws from any distribution, the engine of Monte Carlo simulation.
  • A geometric Brownian motion step: St+Δt=St e(μ−σ2/2)Δt+σΔt zS_{t+\Delta t} = S_t\,e^{(\mu - \sigma^2/2)\Delta t + \sigma\sqrt{\Delta t}\,z}. μ sets the drift of the paths, σ how widely they fan out.
  • Normal N(μ,σ2)N(\mu, \sigma^2): symmetric, mean = median = mode, skewness 0, kurtosis 3 (excess kurtosis 0), fully described by μ and σ.
  • About 68% of outcomes lie within ±1σ of the mean and about 95% within ±2σ.

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