Let N (x) = 192x(0.5 − x)2 for 0 ≤ x ≤ 0.5 and N (x) = 0 for all
other values of x. N (x) is the probability that a soil sample has
a nitrogen concentration of x grams of nitrogen per kilogram of
soil. Let X be the random variable whose value is the nitrogen
concentration in a soil sample. N (x) is the probability density
function for X.
• Show that N (x) is a probability density function.
• Draw the graph of N (x) and explain the meaning of ∫ 0.2
0.1 N (x)dx
in relation to the graph.
• Compute P (0.1 < X < 0.2) and write a sentence explain-
ing the meaning of the value you obtain.
• Determine P (0.3 < X) and explain the meaning of this
value.
• Draw a picture depicting the values of the previous two
parts.
• Compute the expected value of X and explain its meaning
in this context.