Problem 1. Patients infected with a new disease called Chucklepox exhibit elevated levels
of the enzyme Gigglezyme in their bloodstream. If the random variable X represents the
concentration of this enzyme, scientists have found that:
• For healthy patients, X is approximately $N(1, 0.01^2)$ distributed.
• For Chucklepox patients, X is approximately $N(2, 0.03^2)$ distributed.
Scientists design the following diagnostic test for Chucklepox:
• If Gigglezyme levels are more than $t$ units, the test is declared to be positive, suggesting
that the patient has Chucklepox.
• If the Gigglezyme levels are less than $t$ units, the test is declared to be negative,
suggesting that the patient is healthy.
1. (5 points) Compute the probability of a false positive, that is the chance the test is
positive for a healthy patient. Your answer should be a function of $t$, expressed using
$\Phi$, the CDF of the standard Gaussian distribution.
2. (5 points) Compute the probability of a false negative, that is the chance the test is
negative for a sick patient. Your answer should be a function of $t$, expressed using $\Phi$,
the CDF of the standard Gaussian distribution.
3. (10 points) Find the value of $t$ which minimizes the sum of the false positive probability
and the false negative probability.
Hint. For part (3), exploit the fact that the variance of X for both healthy and sick
patients is quite small to make suitable approximations that avoid messy calculations.
