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 \[k\ (V\ -\ v_d) if\ V\ >\ v_d.\ Here\ v_d\] 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 exp

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 \[k\ (V\ -\ v_d) if\ V\ >\ v_d.\ Here\ v_d\] 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 exp

Question
Exponential models
asked 2021-02-13
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 \(\displaystyle{\left[{V}\ \leq\ {v}_{{d}}\ \right]}\) and will \(\displaystyle{\left[{k}\ {\left({V}\ -\ {v}_{{d}}\right)}{\quad\text{if}\quad}\ {V}\ {>}\ {v}_{{d}}.\ {H}{e}{r}{e}\ {v}_{{d}}\right]}\) 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 \(\displaystyle{\left[\lambda\ {f}{\quad\text{or}\quad}\ {V}.\right]}\)
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 \(\displaystyle{w}{>}{0}\).]
b. What is the pdf of W for \(\displaystyle{w}{>}{0}\)? Use this to obtain an expression for the expected value of runoff volume.

Answers (1)

2021-02-14

a) It is given that random variable V denotes the rainfall volume and has exponential distribution with parameter \(\displaystyle{\left[\lambda\right]}\), then using the standard results derived in the book, it's pdf is given as:
\(f_\upsilon(\upsilon)\ =\ \begin{cases}0 & \upsilon < 0\\\lambda\ \cdot\ -e^{-\lambda \upsilon} & \upsilon \geq 0\end{cases}\)
And it's cdf is given as:
\(F_\upsilon(\upsilon)\ =\ \begin{cases}0 & \upsilon < 0\\1\ -e^{-\lambda \upsilon} & \upsilon \geq 0\end{cases}\)
It is also given that rv W denotes runoff volume and it's value depends on V as follows:
\(W\ =\ \begin{cases}0 & V \leq\ \upsilon_d\\k(V\ -\ \upsilon_d) & V\ > \upsilon_d\end{cases}\)
Let us denote cdf of W as \(\displaystyle{\left[{F}_{{w}}{\left({w}\right)}.\right]}\)
As we can see that W can never be less tan zero, hence
\(\displaystyle{\left[{P}{\left({W}\ {<}\ {0}\right)}\ =\ {0}\right]}\)
and using the definition of cdf, we can say that:
\(\displaystyle{\left[{F}_{{w}}{\left({w}\right)}\ =\ {0},\ {f}{\quad\text{or}\quad}\ {w}\ {<}\ {0}\right]}\)
As given in the note, \(\displaystyle{\left[{F}_{{w}}{\left({0}\right)}\ \ne{q}\ {0}\right]}\)
Which means \(\displaystyle{\left[{F}_{{w}}{\left({w}\right)}\right]}\) is not continuous for all values of w. Hence
\(\displaystyle{\left[{F}_{{w}}{\left({0}\right)}\ {P}{\left({W}\right)}\ \leq\ {0}\right.}\)
\(\displaystyle=\ {P}{\left({W}\ =\ {0}\right)}\ +\ {P}{\left({W}\ {<}\ {0}\right)}\)
\(\displaystyle=\ {P}{\left({W}\ =\ {0}\right)}\)
\(\displaystyle=\ {P}{\left({V}\ \leq\ \upsilon_{{d}}\right)}\)
\(\displaystyle=\ {F}_{\upsilon}{\left(\upsilon_{{d}}\right)}\)
\(\displaystyle=\ {1}\ -\ {e}^{{-\lambda\upsilon_{{d}}}}{]}\)
Similiarly for \(\displaystyle{\left[{w}\ {>}\ {0}\right.}\)
\(\displaystyle{F}_{{w}}{\left({w}\right)}\ =\ {P}{\left({W}\ \leq\ {w}\right)}\)
\(\displaystyle=\ {P}{\left({W}\ {<}\ {0}\right)}\ +\ {P}{\left({w}\ =\ {0}\right)}\ +\ {P}{\left({0}\ \leq\ {W}\ {<}\ {w}\right)}\)
\(\displaystyle=\ {0}\ +\ {\left[{1}\ -\ {e}^{{-\lambda\upsilon_{{d}}}}\right]}\ +\ {P}\ {\left({0}\ \leq\ {k}{\left({V}\ -\ \upsilon_{{d}}\right)}\ \leq{w}\right)}\)
\(\displaystyle=\ {\left[{1}\ -\ {e}^{{-\lambda\upsilon_{{d}}}}\right]}\ +\ {P}\ {\left(\upsilon_{{d}}\ \leq\ {V}\ \leq\ \upsilon_{{d}}\ +\ {\frac{{{w}}}{{{k}}}}\right)}\)
\(\displaystyle=\ {\left[{1}\ -\ {e}^{{-\lambda\upsilon_{{d}}}}\right]}\ +\ {\left[{F}_{\upsilon}\ {\left(\upsilon_{{d}}\ +\ {\frac{{{w}}}{{{k}}}}\right)}\ -\ {F}_{\upsilon}{\left(\upsilon_{{d}}\right)}\right]}\)
\(\displaystyle=\ {\left[{1}\ -\ {e}^{{-\lambda\upsilon_{{d}}}}\right]}\ +\ {\left[{\left({1}\ -\ {\exp{{\left(-\lambda\upsilon_{{d}}\ +\ {\frac{{-\lambda{w}}}{{{k}}}}\right)}}}\right)}\ -\ {\left({1}\ -\ {e}^{{-\lambda\upsilon_{{d}}}}\right)}\right]}\)
\(\displaystyle=\ {1}\ -\ {\exp{\ }}{\left(-\lambda\upsilon_{{d}}\ +\ {\frac{{-\lambda{w}}}{{{k}}}}\right)}{]}\)
Finally \(\displaystyle{\left[{F}_{{w}}{\left({w}\right)}\right]}\) can be written as:
\(F_w(w)\ =\ \begin{cases}0 & w < 0\\1\ -\ exp({-\lambda \upsilon_d + \frac{-\lambda w}{k}} & w \geq 0\end{cases}\)
Proposition: Let Z be a continuous rv with cdf \(\displaystyle{\left[\phi{\left({z}\right)}.\right]}\) Then for any number a,
\(\displaystyle{\left[{P}{\left({Z}\ {>}\ \alpha\right)}\ =\ {1}\ -\ \phi{\left(\alpha\right)}\right.}\)
\(\displaystyle{P}{\left({Z}\ {<}\ \alpha\right)}\ =\ \phi{\left(\alpha\right)}{]}\)
(b)
To derive pdf of W from it's cdf, we recall following proposition:
Proposition: If X is a continuous rv with pdf f(x) and cdf F(x), then at every x at which the derative F'(x) exist,
\(\displaystyle{\left[{f{{\left({w}\right)}}}{\left({w}\right)}\ =\ {\frac{{{d}}}{{{d}{w}}}}\ {\left({1}\ -\ {\exp{{\left(-\lambda\upsilon_{{d}}+{\frac{{-\lambda{w}}}{{{k}}}}\right)}}}\right)}\right.}\)
\(\displaystyle={\frac{{{d}}}{{{d}{w}}}}\ {\left({1}\ -\ {e}^{{-\lambda\upsilon_{{d}}}}\cdot{\left({\frac{{-\lambda\omega}}{{{k}}}}\right)}\right)}\)
\(\displaystyle{{f}_{{w}}{\left({w}\right)}}\ =\ {\frac{{\lambda}}{{{k}}}}\ \cdot\ {e}^{{-\lambda\upsilon_{{d}}}}\ \cdot\ {\exp{\ }}{\left({\frac{{-\lambda{w}}}{{{k}}}}\right)}{]}\)
Overall pdf can be written as:
\(f_{w}(w)=\begin{cases}0 & w<0\\1-e^{-\lambda v_{d}} & w=0 \\ \frac{\lambda}{k}\cdot e^{-\lambda v_{d}}\cdot exp(\frac{-\lambda w}{k}) & w>0\end{cases}\)
Now we recall the definition of expexted value of a pdf:
Definition: The expected or mean value of a continuous rv X with pdf \({f{{\left({x}\right)}}}\ {i}{s}\)
\(\displaystyle{E}{\left({X}\right)}\ =\ {\int_{{-\infty}}^{{\infty}}}{x}\ \cdot\ {f{{\left({x}\right)}}}\ \cdot\ {\left.{d}{x}\right.}\)
Using this, we can write \(\displaystyle{\left[{E}{\left({W}\right)}\right]}\) as:
\(\displaystyle{\left[{E}{\left({X}\right)}\ =\ {\int_{{-\infty}}^{{\infty}}}\omega\ \cdot\ {{f}_{{\omega}}{\left(\omega\right)}}\ \cdot\ {d}\omega\right.}\)
\(\displaystyle=\ {\int_{{{0}}}^{{\infty}}}\ {w}\ \cdot\ {\frac{{\lambda}}{{{k}}}}\ \cdot\ {e}^{{-\lambda\upsilon_{{d}}}}\ \cdot\ {\exp{{\left({\frac{{-\lambda{w}}}{{{k}}}}\right)}}}{d}{w}\)
\(\displaystyle=\ {\frac{{\lambda\ \cdot\ {e}^{{-\lambda\upsilon_{{d}}}}}}{{{k}}}}\ {\int_{{{0}}}^{{\infty}}}\ {w}\ \cdot\ {\exp{\ }}{\left({\frac{{-\lambda\omega}}{{{k}}}}{d}{w}\right)}{]}\)
Here we will use integration by parts, which can be summarized as follows:
Integration by parts: If u and v are function of x, then
\(\int_{a}^{b}\ u \upsilon^{\prime} \cdot\ dx\ =\ [u \upsilon]_{a}^{b}\ -\ \int_{a}^{b}\ u^{\prime} \upsilon\ \cdot\ dx\)
In our case, we take \(\displaystyle{\left[{u}\ =\ {w}\right]}{\quad\text{and}\quad}{\left[\upsilon\ =\ {\exp{\ }}{\left({\frac{{-\lambda{w}}}{{{k}}}}\right)}\right]}\)
Then the integral becomes:
\(\displaystyle{\left[{E}{\left({W}\right)}\ =\ {\frac{{\lambda\ \cdot\ {e}^{{-\lambda\upsilon_{{d}}}}}}{{{k}}}}\ {\left({\left[-{w}\ \cdot\ {\frac{{{k}}}{{\lambda}}}\ \cdot\ {\exp{\ }}{{\left({\frac{{-\lambda{w}}}{{{k}}}}\right]}_{{{0}}}^{{\infty}}}\ +\ {\frac{{{k}}}{{\lambda}}}\ \cdot\ {\int_{{{0}}}^{{\infty}}}\ {\exp{{\left({\frac{{-\lambda{w}}}{{{k}}}}\right)}}}{d}{w}\right)}\right.}\right.}\)
\(\displaystyle{E}{\left({W}\right)}\ =\ {\frac{{\lambda\ \cdot\ {e}^{{-\lambda\upsilon_{{d}}}}}}{{{k}}}}\ {\left({{\left[-{w}\ \cdot\ {\frac{{{k}}}{{\lambda}}}\ \cdot\ {\exp{\ }}{\left({\frac{{-\lambda{w}}}{{{k}}}}\right)}\right]}_{{{0}}}^{{\infty}}}\ -\ {\frac{{{k}^{{2}}}}{{\lambda^{{2}}}}}\ \cdot\ {{\left[{\exp{{\left({\frac{{-\lambda{w}}}{{{k}}}}\right)}}}\right]}_{{{0}}}^{{\infty}}}\right)}\)
\(\displaystyle{E}{\left({W}\right)}\ =\ {\frac{{\lambda\ \cdot\ {e}^{{-\lambda\upsilon_{{d}}}}}}{{{k}}}}\ {\left({\left[{0}\ -\ {0}\right]}\ -{\frac{{{k}^{{2}}}}{{\lambda^{{2}}}}}\ \cdot\ {\left[{0}\ -\ {1}\right]}\right)}\)
\(\displaystyle{E}{\left({W}\right)}\ =\ {\frac{{\lambda\ \cdot\ {e}^{{-\lambda\upsilon_{{d}}}}}}{{{k}}}}\ {\left({\frac{{{k}^{{2}}}}{{\lambda^{{2}}}}}\right)}\)
\(\displaystyle{E}{\left({W}\right)}\ =\ {\frac{{\lambda\ \cdot\ {e}^{{-\lambda\upsilon_{{d}}}}}}{{\lambda}}}\)

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