. Let V denote rainfall volume and Wdenote runoff volume (both in mm). According to the article
“Runoff Quality Analysis of Urban Catchments with Analytical Probability Models”
(J. of Water Resource Planning and Management, 2006: 4–14), the runoff volume will be 0 if
V _ vd and will be k(V _ vd) if V > vd. Here vd is the volume of depression storage (a constant),
and k (also a constant) is the runoff coefficient. The cited article proposes an exponential
distribution with parameter λ for V.
(a) Obtain an expression for the cdf of W. [Note: W is neither purely continuous nor purely
discrete; instead it has a “mixed” distribution with a discrete component at 0 and is
continuous for values w > 0.]
(b) What is the pdf of W for w > 0? Use this to obtain an expression for the expected value of
runoff volume.