# Distributions for Disease Exposure Levels

A rare disease has just broken out. Doctor Johnson is trying to help, but while treating patients he might expose himself to the disease. He takes many precautions, but he doesn't know how much he's been exposed. Let the number x represent Doctor Johnson's exposure level. He doesn't know it for sure, but he assigns a uniform distribution to it(between 0 and 1). He also assigns a probability

{Doctor Johnson gets disease | x} = x^2

Doctor Johnson has just come down with the disease. What is his new distribution for his exposure level?

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#### Solution Preview

Let's let D be the random variable representing Doctor Johnson getting the disease. It's event list is { Doctor Johnson gets disease, Doctor Johnson doesn't get disease). We will represent these in shorthand as 1 and 0, for gets disease and doesn't get disease, respectively. The probability density function, pD, is not given and must be derived.

Let X be the random variable describing the level of exposure. The possible values of this variable are in the continuous range 0 to 1. The probability density function pX is given as uniform, ...

#### Solution Summary

The solution addresses a rare disease has just broken out. Doctor Johnson is trying to help, but while treating patients he might expose himself to the disease. He takes many precautions, but he doesn't know how much he's been exposed. Let the number x represent Doctor Johnson's exposure level. He doesn't know it for sure, but he assigns a uniform distribution to it(between 0 and 1). He also assigns a probability

{Doctor Johnson gets disease | x} = x^2

Doctor Johnson has just come down with the disease. What is his new distribution for his exposure level?