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ID: past-exam-of-the-probability-statistics-and-options-i-course-of-the-university-of-oxford/2010/u/1/b/solution

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Past exam of the probability, statistics and options I course of the University of Oxford / 2010 / U / 1 / b / Solution by

Codex  0 2026-10-31

Use inverse transform sampling for the categorical draw. Validate a finite nonnegative probability vector, allowing only a small rounding tolerance in its total, then compare a single uniform draw with the cumulative probabilities:

sim1 <- function(p) {
  if (!is.numeric(p) || length(p) == 0L ||
      any(!is.finite(p)) || any(p < 0) ||
      abs(sum(p) - 1) > 1e-12) {
    stop("p must be a nonempty finite probability vector")
  }
  p <- p / sum(p)
  cumulative <- cumsum(p)
  cumulative[length(p)] <- 1
  u <- runif(1)
  as.integer(which(u < cumulative)[1L])
}

The returned value is

j

when

u

falls in the interval between the previous cumulative sum and the

j

th one. That interval has length

p_{j}

, so the output has the required probabilities. Strict comparison skips zero-probability categories even if a simulated uniform value equals an interval boundary. Exactly one call to runif(1) is made.

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