Skip to main content

Answer

Prerequisites​

  • Exponential Growth/Decay
  • Likelihood Ratio

Step-by-Step Derivation​

From part (c), the decision rule is to guess heads if: (1−θ1)kθ1n−kα>θ2k(1−θ2)n−k(1−α)(1 - \theta_1)^k \theta_1^{n-k} \alpha > \theta_2^k (1 - \theta_2)^{n-k} (1 - \alpha)

We are given two conditions:

  1. Symmetric error rates: θ1=θ2=θ\theta_1 = \theta_2 = \theta
  2. The report sequence is all heads: This means k=nk = n and n−k=0n - k = 0.

Substituting these conditions into the rule: (1−θ)nθ0α>θn(1−θ)0(1−α)(1 - \theta)^n \theta^0 \alpha > \theta^n (1 - \theta)^0 (1 - \alpha) (1−θ)nα>θn(1−α)(1 - \theta)^n \alpha > \theta^n (1 - \alpha)

Assuming θ>0\theta > 0, we can rearrange this to form a likelihood ratio: (1−θθ)n>1−αα\left(\frac{1 - \theta}{\theta}\right)^n > \frac{1 - \alpha}{\alpha}

This is the simplified decision rule for guessing heads under these specific conditions.