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

illusiia 2021-02-13 Answered
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  vd ] and will [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.
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Expert Answer

Cristiano Sears
Answered 2021-02-14 Author has 96 answers

a) It is given that random variable V denotes the rainfall volume and has exponential distribution with parameter [λ], then using the standard results derived in the book, it's pdf is given as:
fυ(υ) = {0υ<0λ  eλυυ0
And it's cdf is given as:
Fυ(υ) = {0υ<01 eλυυ0
It is also given that rv W denotes runoff volume and it's value depends on V as follows:
W = {0V υdk(V  υd)V >υd
Let us denote cdf of W as [Fw(w).]
As we can see that W can never be less tan zero, hence
[P(W < 0) = 0]
and using the definition of cdf, we can say that:
[Fw(w) = 0, for w < 0]
As given in the note, [Fw(0) q 0]
Which means [Fw(w)] is not continuous for all values of w. Hence
[Fw(0) P(W)  0
= P(W = 0) + P(W < 0)
= P(W = 0)
= P(V  υd)
= Fυ(υd)
= 1  eλυd]
Similiarly for [w > 0
Fw(w) = P(W  w)
= P(W < 0) + P(w = 0) + P(0  W < w)
= 0 + [1  eλυd] + P (0  k(V  υd) w)
= [1  eλυd] + P (υd  V  υd + wk)
= [1  eλυd] + [Fυ (υd + wk)  Fυ(υd)]

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