A car panel is spray-painted by a machine, and the technicians are particularly interested in the thickness of the resulting paint layer. Suppose that the random variable X measures the thickness of the paint in millimeters at a randomly chosen point on a randomly chosen car panel, and that X takes values between 0.125 and 0.5 mm with a probability density function of
f(x) = A[0.5 - (x - 0.25)^2]
for 0.125 <= x <= 0.5 and f(x) = 0 elsewhere.
a. Find the value of A that make f(x) a valid pdf.
b. Construct the cumulative distribution function.
c. Find the probability that the paint thickness at particular point is less than 0.2 mm, larger than 0.35mm, and between 0.35 and 0.2 mm respectively.
d. What is the expected paint thickness?
e. What is the variance and standard deviation of paint thickness?