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 and will be if . Here 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 .]

b. What is the pdf of W for ? Use this to obtain an expression for the expected value of runoff volume.