gistfile1.txt

First, given the PDF we determine the corresponding CDF. While the following notation is mathematically not entirely correct, it simplifies the illustraion:

f1(x) = x/144 if 0 < x < 12
f2(x) = (24-x)/144 if 12 < x < 24

F1(x) = x^2/288 + c if 0 < x < 12
F2(x) = x/6-x^2/288 + c if 12 < x < 24

Next, we find the probabilities that:

a) X is between 1 and 10
P(1 < X < 10) = F1(10) - F1(1) = 10^2/288 - 1^2/288 = 11/32 ~= 0.34

b) X is between 11 and 16
P(11 < X < 16) = [F2(16) - F2(12)] + [F1(12) - F1(11)]  = [(16/6-16^2/288) - (12/6-12^2/288)] + [(12^2/288) - (11^2/288)] = 103/288 ~= 0.36

c) X is less than 15
P(X < 15) = [F2(15) - F2(12)] + [F1(12) - F1(0)]  = [(15/6-15^2/288) - (12/6-12^2/288)] + [(12^2/288) - (0^2/288)] = 23/33 ~= 0.72

d) X is greater than 10
P(X > 10) = 1 - P(X < 10) = 1 - F1(10) = 1 - [10^2/288] = 47/72 = 0.65