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 &gt; 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&gt;0.] b. What is the pdf of W for w&gt;0? Use this to obtain an expression for the expected value of runoff volume.

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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 &amp;gt; 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&amp;gt;0.]  b. What is the pdf of W for w&amp;gt;0? Use this to obtain an expression for the expected value of runoff volume.

Cristiano Sears · Accepted Answer

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&#039;s pdf is given as: fυ(υ) = {0υ&amp;lt;0λ ⋅ −e−λυυ≥0 And it&#039;s cdf is given as: Fυ(υ) = {0υ&amp;lt;01 −e−λυυ≥0 It is also given that rv W denotes runoff volume and it&#039;s value depends on V as follows: W = {0V≤ υdk(V − υd)V &amp;gt;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 &amp;lt; 0) = 0] and using the definition of cdf, we can say that: [Fw(w) = 0, for w &amp;lt; 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 &amp;lt; 0) = P(W = 0) = P(V ≤ υd) = Fυ(υd) = 1 − e−λυd] Similiarly for [w &amp;gt; 0 Fw(w) = P(W ≤ w) = P(W &amp;lt; 0) + P(w = 0) + P(0 ≤ W &amp;lt; w) = 0 + [1 − e−λυd] + P (0 ≤ k(V − υd) ≤w) = [1 − e−λυd] + P (υd ≤ V ≤ υd + wk) = [1 − e−λυd] + [Fυ (υd + wk) − Fυ(υd)]

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

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