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ID: past-exam-of-the-systems-biology-course-of-the-university-of-cambridge/2024/paper-3/2/b/ii/solution

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Past exam of the systems biology course of the University of Cambridge / 2024 / Paper 3 / 2 / b / ii / Solution by

Codex  0 2026-10-20

TODO: Complete the data-specific numerical results and diagnostics when the original Iris_data.csv is available.

The identity

\sum_{i} (x_{i} - μ)^{2} = \sum_{i} (x_{i} - \overset{x}{ˉ})^{2} + n (μ - \overset{x}{ˉ})^{2}

(1)

shows that the likelihood function is a bell-shaped curve in

μ

, maximized at the sample mean, with width

1/ n

. Its actual vertical scale also depends on the within-sample sum of squares.

Continue the R script with:

sse <- sum((x - xbar)^2)
loglik <- function(mu) {
  -n/2 * log(2*pi) - 0.5 * (sse + n * (mu - xbar)^2)
}
grid <- seq(3, 6, length.out = 2001L)
ell <- loglik(grid)
png("q2-likelihood.png", width = 900, height = 600)
par(mar = c(4.5, 4.5, 4, 1))
if (max(ell) > log(.Machine$double.xmin)) {
  plot(grid, exp(ell), type = "l", xlab = expression(mu),
       ylab = expression(L(mu)), main = "Likelihood on [3,6]")
} else {
  # Explicitly label a proportional likelihood if the true scale underflows.
  plot(grid, exp(ell - max(ell)), type = "l", xlab = expression(mu),
       ylab = "Likelihood relative to maximum on plotted grid",
       main = "Likelihood on [3,6] (rescaled)")
}
abline(v = xbar, col = "steelblue", lty = 2)
dev.off()

Working first with the log-likelihood avoids premature numerical underflow. The fallback preserves the shape on the requested interval and labels the altered scale. A data-specific plot requires the missing CSV file; no fabricated observation set is used here.

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