$13.$ The normalized difference vegetation index $($NDVI$)$ is commonly used to classify land cover using remote sensing data. Hypothetically, say that NDVI follows a Beta$(25,10)$ distribution for pixels in a rain forest, and a Beta$(10,15)$ distribution for pixels in a deforested area now used for agriculture. Assuming about $10\%$ of the rain forest has been deforested, your objective is to build a rule to classify individual pixels as deforested based on their NDVI.

$($a$)$ Plot the PDF of NDVI for forested and deforested pixels, and the marginal distribution of NDVI averaging over categories.

$($b$)$ Give an expression for the probability that a pixel is deforested given its NDVI value, and plot this probability by NDVI.

$($c$)$ You will classify a pixel as deforested if you are at least $90\%$ sure it is deforested. Following this rule, give the range of NDVI that will lead to a pixel being classified as deforested.