128. Let V denote rainfall volume and W denote 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  d and will be k(V  d ) if V d . Here d 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  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.