The bulk density of soil is defined as the mass of dry solids per unit volume. A high bulk density implies a compact soil with few pores. Bulk density is an important factor in influencing root development, seedling emergence, and aeration. Let X denote the bulk density of Pima clay loam . Studies show that X is normally distributed with μ =1.5 andσ=0.2g/cm³.

a)What is the density for X? Sketch a graph of the density function. Indicate on the graph the probability that X lies between 1.1 and 1.9. Find this probability.

b)Find the probability that a randomly selected sample of Pimaclay loam will have a bulk density less than 0.9g/cm³.

c)Would you be surprised if a randomly selected sample of this type of soil has a bulk density in excess of 2.0g/cm³. Explain, based on the probability of this occurring.

d)What point has the property that only 10% of the soil samples have bulk density this high or higher?

e)What is the moment generating function for X?