Compute true positive rate from only predicted probability and actual probability (Binary LDA classifier)

Question

I'm sort of stuck on this question and I can't find a similar problem online.

Consider the following to be given:
1.input x is 1D, output y is binary {0,1}
2.marginal probability of y is $\pi_y=P(y)$
3.the conditional distribution of $x|y$ is Gaussian normal, $\mu_y$ depends on binary y, variance $\sigma^2$ does not depend on y, so:

4.The probability of $P(y|x)$ is:

5.The LDA classifier for the case $\mu_1>\mu_0$ is:

Now I know that $\sigma^2=0.25$, $\pi_1=60\%$, $\pi_0=40\%$, $\mu_1=0.5$, $\mu_0=-0.5$; sample size is not known.
How can I calculate the tpr(true positive rate) in this case?

My current thought is,as reflected from title, from assumption 5, I can plug in numbers to see that the predicted label is 1 when $x>=0.25log(1.5)$ and take the normal density integral on x from $0.25log(1.5)$ to infinity; the result will be $P(\hat{y}=1)$ (subquestion: is the result $P(\hat{y}=1)$ or $P(\hat{y}=1|x)$). $P(y=1)$ is given, which is $\pi_1=60\%$.

From here, how do I continue do find the true positive rate? Thank you.

Answer 1

TPR is choosing $\hat{y}=1$ when $y$ is really $1$, i.e. formally $$P(\hat{y}=1|y=1)=P(X\geq \tau|y=1)$$

$p(x|y)$ is given here, and needs to be integrated from $\tau$ to $\infty$, where $\tau=0.25\log 1.5$. So, your approach is correct. However, since you cannot integrate the normal PDF, we'll need to convert it to standard normal and use the CDF table.

$$P(X\geq \tau|y=1)=P\left(Z\geq\frac{\tau-\mu_1}{\sigma}\right)\approx P(Z\geq -0.8)=1-\Phi(-0.8)=\Phi(0.8)\approx 0.7881$$

Note: in bullet point 3, you say $y|x$ is normal but it should be $x|y$.