I am dealing with a problem written as follows:-
Suppose that 10% of the people in a certain population have the eye disease glaucoma. For persons who have glaucoma measurements of eye pressure X will have normally distributed X with a mean of 25 and a variance of 1. For persons whithout glaucoma the pressure X is normally distributed with a mean of 20 and a variance of 1. Suppose a person is selected at random from the population and eye pressure X is measured. Determine the probability that the person has glaucoma given that X = a.
First I note that the probability if having glaucoma + probability of not having it = 1, even at a pressure X = 23, say which is several standard deviations away from each of the means 20 and 25.
If I write the Gaussian, just for brevity here, with mean 20 as f20(X) and the one with mean 25 as f25(X) then I think the probability of being in the set "has glaucoma" might be expressed as:-
0.1 f25(a)/[0.9f20(a) + 0.1f25(a)]
If this is correct, then I'm having trouble expressing myself why this is correct. It seems to have the right behaviour, making a rapid transition between 0 and 1 as we go above the intermediate value a = 22.5
Comments would be helpful, perhaps I've gotten this entirely wrong.