This problem is for a probability review but I am not sure how to go about solving it:
Population 1 has 5 times as many people as population 2. In population 1, a certain biometric variable has normal distribution with mean 90 and standard deviation 15. In population 2, the same biometric variable has normal distribution with mean 60 and standard deviation 15. If a randomly chosen person from either population is measured at 72, what is the conditional probability that the person belongs to population 2?
Attempt at setting up the problem:
The total population: $$P_t=P_1+P_2$$ $$\mu_1 = 90, \sigma_1 = 15$$ $$\mu_2 = 60, \sigma_2 = 15$$
I think I may need to find a joint distribution, and I think that they are independent.
Since the population P1 is 5 times greater than P2, to make the pdf's proportional: $$\sigma_1' = \frac{\sigma_1}{\sqrt5}$$ $$\sigma=\sigma_1=\sigma_2$$
And the joint pdf would be: $$f_{XY}(x,y)=\frac{1}{\sigma\sqrt{2\pi/5}}exp(\frac{-5(x-\mu_1)}{2\sigma^2})\frac{1}{\sigma\sqrt{2\pi}}exp(\frac{-(y-\mu_2)}{2\sigma^2})$$
I think the next step would be to do a condition probability to find
$$P(Y|X or Y=72)$$
But I am not sure how to solve this. I am also not sure if finding the standard deviation for the proportional probability was the right step.
If anyone could help me solve this that would be great! Thank you.