We have developed an optical imaging system for single-cell inspection. Each cell can be either normal or cancerous. Through extensive studies, we find that the two types of cells have different physiological characteristics. In particular, it is possible to dierentiate them in terms of the cell diameter.

For a normal cell, the mean diameter is1=3m, with standard deviation1=0.4m.

For a cancerous cell, the mean diameter is2=4.5m, with standard deviation2=0.6m.

In an experiment, we have a mixture containing80% of normal and20% of cancerous cells. We now image a specific cell and find that it has a diameter of3.7m.

We assume that both normal and cancerous cells follow Gaussian distributions, with mean and standard deviation as stated above.

(a) What is the posterior probability that the cell is normal?

(b) What is the posterior probability that the cell is cancerous?

(c) Using MAP(maximum a posterior), would you conclude the cell is normal or cancerous?