# Let V denote rainfall volume and W denote runoff volume (both in mm). According to the article “Runoff Quality Analysis of Urban Catchments with Analy

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 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
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.
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Cristiano Sears

a) It is given that random variable V denotes the rainfall volume and has exponential distribution with parameter $\left[\lambda \right]$, then using the standard results derived in the book, it's pdf is given as:

And it's cdf is given as:

It is also given that rv W denotes runoff volume and it's value depends on V as follows:

Let us denote cdf of W as $\left[{F}_{w}\left(w\right).\right]$
As we can see that W can never be less tan zero, hence

and using the definition of cdf, we can say that:

As given in the note,
Which means $\left[{F}_{w}\left(w\right)\right]$ is not continuous for all values of w. Hence

Similiarly for