91Ó°ÊÓ

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 \(V \leq v_{d}\) and will be \(k\left(V-v_{d}\right)\) if \(V>v_{d}\). Here \(v_{d}\) 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 \(\lambda\) 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.

Step by step solution

01

Define the CDF for Runoff Volume W

The cumulative distribution function (CDF) for runoff volume, \( W \), must account for the probability that \( W = 0 \) and the probability that \( W > 0 \). When the rainfall volume \( V \) is less than or equal to \( v_d \), all rainfall is absorbed, resulting in no runoff. Therefore, \( P(W = 0) = P(V \leq v_d) \). Given \( V \) follows an exponential distribution with parameter \( \lambda \), the CDF for \( V \) is \( 1 - e^{-\lambda V} \). Thus, \( P(W = 0) = 1 - e^{-\lambda v_d} \). If \( V > v_d \), the runoff volume \( W = k(V - v_d) \). So for \( w > 0 \), \( P(W \leq w) = P(V \leq v_d + \frac{w}{k}) \), resulting in \( 1 - e^{-\lambda (v_d + \frac{w}{k})} \). Combine these to get the CDF of \( W \): \[F_W(w) = \begin{cases} 1 - e^{-\lambda v_d}, & w = 0, \1 - e^{-\lambda (v_d + \frac{w}{k})}, & w > 0.\end{cases}\]

02

Obtain the PDF for W where w > 0

For \( w > 0 \), \( W \) is continuous and we can find the PDF by differentiating the continuous part of the CDF with respect to \( w \). The continuous part of the CDF is \( 1 - e^{-\lambda (v_d + \frac{w}{k})} \). Differentiating this with respect to \( w \) involves the chain rule:\[f_W(w) = \frac{d}{dw}[1 - e^{-\lambda (v_d + \frac{w}{k})}] = \lambda e^{-\lambda (v_d + \frac{w}{k})} \cdot \frac{1}{k} = \frac{\lambda}{k} e^{-\lambda (v_d + \frac{w}{k})}\]

03

Calculate the Expected Value of Continuous Part of Runoff Volume

The expected value of the continuous part of runoff volume \( W \) can be calculated using the PDF derived:\[E[W] = \int_{0}^{\infty} w \cdot \frac{\lambda}{k} e^{-\lambda (v_d + \frac{w}{k})} \, dw\]By substituting \( u = \frac{w}{k} \), the integral changes to:\[E[W] = \int_{v_d}^{\infty} k(u - v_d) \cdot \lambda e^{-\lambda u} \, du\]Evaluating this integral gives:\[E[W] = ke^{-\lambda v_d}\]

04

(Optional): Consider Total Expectation from Mixed Distribution

Since there is a probability \( 1 - e^{-\lambda v_d} \) of \( W = 0 \) and the remaining probability distribution is continuous, the total expected value is a weighted average considering both the probability of no runoff and runoff. Since the expectation for \( W = 0 \) is simply zero:\[E[W_{total}] = (1 - e^{-\lambda v_d}) \cdot 0 + e^{-\lambda v_d} \cdot ke^{-\lambda v_d} = ke^{-\lambda v_d}\]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Runoff Volume

Runoff volume represents the amount of excess water flowing off a surface after rainfall has taken place. It is crucial in water resource management, particularly in urban areas where excess runoff can lead to flooding.
In our exercise, runoff volume, denoted as \( W \), is influenced by two primary parameters: the depression storage volume \( v_d \) and the runoff coefficient \( k \).

• The depression storage volume \( v_d \) is the amount of precipitation that the surface can absorb before any runoff occurs.
• The runoff coefficient \( k \) dictates the proportion of rainfall that converts into runoff once \( v_d \) is exceeded.

When the rainfall volume \( V \) is less than or equal to \( v_d \), no runoff occurs, and therefore, \( W = 0 \). For rainfall volumes exceeding \( v_d \), runoff volume is calculated as \( W = k(V - v_d) \). This relationship establishes \( W \) as a mixed distribution, reflecting both discrete and continuous characteristics depending on the rainfall context.

Exponential Distribution

The rainfall volume \( V \) follows an exponential distribution with a parameter \( \lambda \), a common model for describing the time or space between random events.
This distribution is favored because:

• It assumes that events (such as rainfalls) occur continuously and independently over time.
• The exponential distribution is defined only by its rate parameter \( \lambda \), simplifying calculations.

The probability density function (PDF) for an exponentially distributed variable is \( f_V(v) = \lambda e^{-\lambda v} \), describing the relative likelihood that \( V \) takes a specific value.
The cumulative distribution function (CDF), which is \( F_V(v) = 1 - e^{-\lambda v} \), gives the probability that \( V \) is less than or equal to a specific value. This illustrates how much rainfall is likely to be captured as runoff if it exceeds the depression storage volume \( v_d \). Understanding this distribution is essential for determining subsequent runoff volume characteristics.

Cumulative Distribution Function (CDF)

The cumulative distribution function (CDF) of a random variable provides a complete overview of its distribution, showing the probability of a variable being less than or equal to a particular value. For upfront rainfall volume \( V \), which is exponentially distributed, the CDF \( F_V(v) = 1 - e^{-\lambda v} \) describes how likely it is for \( V \) to be less than or equal to any given volume.
When considering runoff volume \( W \), the CDF needs to account for both the cases where the runoff is zero and where it exceeds zero. Thus, the CDF for \( W \) involves two expressions:

• \( P(W = 0) = 1 - e^{-\lambda v_d} \)
• For \( w > 0 \), \( P(W \leq w) = 1 - e^{-\lambda (v_d + \frac{w}{k})} \)

This mixed distribution reflects that \( W \) can jump from a discrete zero state to continuous values based on rainfall exceeding \( v_d \). The mixed nature is crucial for accurate runoff predictions in urban planning and resource management.

Probability Density Function (PDF)

The probability density function (PDF) provides insights into the relative likelihood of a continuous random variable assuming a particular value.
In the case of runoff volume \( W \) for \( w > 0 \), the PDF is integral for calculating expected values and other statistical measures.
Differentiation of the continuous part of the CDF \( 1 - e^{-\lambda (v_d + \frac{w}{k})} \) gives us the PDF for \( W \):

• For \( w > 0 \), the PDF is \( f_W(w) = \frac{\lambda}{k} e^{-\lambda (v_d + \frac{w}{k})} \).

Understanding the PDF is crucial for statistical analyses, such as determining the expected runoff or the probability of certain runoff volumes occurring.
It also facilitates more refined models for predicting water management needs and potential flood risk in urban planning.