# 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
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.

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:
$$\displaystyle{\left[{E}{\left({W}\right)}\ =\ {\frac{{\lambda\ \cdot\ {e}^{{-\lambda\upsilon_{{d}}}}}}{{{k}}}}\ {\left({\left[-{w}\ \cdot\ {\frac{{{k}}}{{\lambda}}}\ \cdot\ {\exp{\ }}{{\left({\frac{{-\lambda{w}}}{{{k}}}}\right]}_{{{0}}}^{{\infty}}}\ +\ {\frac{{{k}}}{{\lambda}}}\ \cdot\ {\int_{{{0}}}^{{\infty}}}\ {\exp{{\left({\frac{{-\lambda{w}}}{{{k}}}}\right)}}}{d}{w}\right)}\right.}\right.}$$
$$\displaystyle{E}{\left({W}\right)}\ =\ {\frac{{\lambda\ \cdot\ {e}^{{-\lambda\upsilon_{{d}}}}}}{{{k}}}}\ {\left({{\left[-{w}\ \cdot\ {\frac{{{k}}}{{\lambda}}}\ \cdot\ {\exp{\ }}{\left({\frac{{-\lambda{w}}}{{{k}}}}\right)}\right]}_{{{0}}}^{{\infty}}}\ -\ {\frac{{{k}^{{2}}}}{{\lambda^{{2}}}}}\ \cdot\ {{\left[{\exp{{\left({\frac{{-\lambda{w}}}{{{k}}}}\right)}}}\right]}_{{{0}}}^{{\infty}}}\right)}$$
$$\displaystyle{E}{\left({W}\right)}\ =\ {\frac{{\lambda\ \cdot\ {e}^{{-\lambda\upsilon_{{d}}}}}}{{{k}}}}\ {\left({\left[{0}\ -\ {0}\right]}\ -{\frac{{{k}^{{2}}}}{{\lambda^{{2}}}}}\ \cdot\ {\left[{0}\ -\ {1}\right]}\right)}$$
$$\displaystyle{E}{\left({W}\right)}\ =\ {\frac{{\lambda\ \cdot\ {e}^{{-\lambda\upsilon_{{d}}}}}}{{{k}}}}\ {\left({\frac{{{k}^{{2}}}}{{\lambda^{{2}}}}}\right)}$$
$$\displaystyle{E}{\left({W}\right)}\ =\ {\frac{{\lambda\ \cdot\ {e}^{{-\lambda\upsilon_{{d}}}}}}{{\lambda}}}$$

### Relevant Questions

As depicted in the applet, Albertine finds herself in a very odd contraption. She sits in a reclining chair, in front of a large, compressed spring. The spring is compressed 5.00 m from its equilibrium position, and a glass sits 19.8m from her outstretched foot.
a)Assuming that Albertine's mass is 60.0kg , what is $$\displaystyle\mu_{{k}}$$, the coefficient of kinetic friction between the chair and the waxed floor? Use $$\displaystyle{g}={9.80}\frac{{m}}{{s}^{{2}}}$$ for the magnitude of the acceleration due to gravity. Assume that the value of k found in Part A has three significant figures. Note that if you did not assume that k has three significant figures, it would be impossible to get three significant figures for $$\displaystyle\mu_{{k}}$$, since the length scale along the bottom of the applet does not allow you to measure distances to that accuracy with different values of k.
The bulk density of soil is defined as the mass of dry solidsper unit bulk volume. A high bulk density implies a compact soilwith few pores. Bulk density is an important factor in influencing root development, seedling emergence, and aeration. Let X denotethe bulk density of Pima clay loam. Studies show that X is normally distributed with $$\displaystyle\mu={1.5}$$ and $$\displaystyle\sigma={0.2}\frac{{g}}{{c}}{m}^{{3}}$$.
(a) What is thedensity for X? Sketch a graph of the density function. Indicate onthis graph the probability that X lies between 1.1 and 1.9. Findthis probability.
(b) Find the probability that arandomly selected sample of Pima clay loam will have bulk densityless than $$\displaystyle{0.9}\frac{{g}}{{c}}{m}^{{3}}$$.
(c) Would you be surprised if a randomly selected sample of this type of soil has a bulkdensity in excess of $$\displaystyle{2.0}\frac{{g}}{{c}}{m}^{{3}}$$? Explain, based on theprobability of this occurring.
(d) What point has the property that only 10% of the soil samples have bulk density this high orhigher?
(e) What is the moment generating function for X?
A box is sliding with a speed of 4.50 m/s on a horizontal surface when, at point P, it encounters a rough section. On the rough section, the coefficient of friction is not constant but starts at .100 at P and increases linerly with distance past P, reaching a value of .600 at 12.5 m past point P. (a) Use the work energy theorem to find how far this box slides before stopping. (b) What is the coefficient of friction at the stopping point? (c) How far would the box have slid iff the friciton coefficient didn't increase, but instead had the constant value of .1?
The dominant form of drag experienced by vehicles (bikes, cars,planes, etc.) at operating speeds is called form drag. Itincreases quadratically with velocity (essentially because theamount of air you run into increase with v and so does the amount of force you must exert on each small volume of air). Thus
$$\displaystyle{F}_{{{d}{r}{u}{g}}}={C}_{{d}}{A}{v}^{{2}}$$
where A is the cross-sectional area of the vehicle and $$\displaystyle{C}_{{d}}$$ is called the coefficient of drag.
Part A:
Consider a vehicle moving with constant velocity $$\displaystyle\vec{{{v}}}$$. Find the power dissipated by form drag.
Express your answer in terms of $$\displaystyle{C}_{{d}},{A},$$ and speed v.
Part B:
A certain car has an engine that provides a maximum power $$\displaystyle{P}_{{0}}$$. Suppose that the maximum speed of thee car, $$\displaystyle{v}_{{0}}$$, is limited by a drag force proportional to the square of the speed (as in the previous part). The car engine is now modified, so that the new power $$\displaystyle{P}_{{1}}$$ is 10 percent greater than the original power ($$\displaystyle{P}_{{1}}={110}\%{P}_{{0}}$$).
Assume the following:
The top speed is limited by air drag.
The magnitude of the force of air drag at these speeds is proportional to the square of the speed.
By what percentage, $$\displaystyle{\frac{{{v}_{{1}}-{v}_{{0}}}}{{{v}_{{0}}}}}$$, is the top speed of the car increased?
Express the percent increase in top speed numerically to two significant figures.
We will now add support for register-memory ALU operations to the classic five-stage RISC pipeline. To offset this increase in complexity, all memory addressing will be restricted to register indirect (i.e., all addresses are simply a value held in a register; no offset or displacement may be added to the register value). For example, the register-memory instruction add x4, x5, (x1) means add the contents of register x5 to the contents of the memory location with address equal to the value in register x1 and put the sum in register x4. Register-register ALU operations are unchanged. The following items apply to the integer RISC pipeline:
a. List a rearranged order of the five traditional stages of the RISC pipeline that will support register-memory operations implemented exclusively by register indirect addressing.
b. Describe what new forwarding paths are needed for the rearranged pipeline by stating the source, destination, and information transferred on each needed new path.
c. For the reordered stages of the RISC pipeline, what new data hazards are created by this addressing mode? Give an instruction sequence illustrating each new hazard.
d. List all of the ways that the RISC pipeline with register-memory ALU operations can have a different instruction count for a given program than the original RISC pipeline. Give a pair of specific instruction sequences, one for the original pipeline and one for the rearranged pipeline, to illustrate each way.
Hint for (d): Give a pair of instruction sequences where the RISC pipeline has “more” instructions than the reg-mem architecture. Also give a pair of instruction sequences where the RISC pipeline has “fewer” instructions than the reg-mem architecture.
The coefficient of linear expansion of copper is 17 x 10-6 K-1. A sheet of copper has a round hole with a radius of 3.0 m cut out of it. If the sheet is heated and undergoes a change in temperature of 80 K, what is the change in the radius of the hole? It decreases by 4.1 mm. It increases by 4.1 mm. It decreases by 8.2 mm. It increases by 8.2 mm. It does not change.
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A. Find the radii of the two "daughter" nuclei of charge+46e.
B. In a simple model for the fission process, immediatelyafter the uranium-236 nucleus has undergone fission the "daughter"nuclei are at rest and just touching. Calculate the kineticenergy that each of the "daughter" nuclei will have when they arevery far apart.
C. In this model the sum of the kinetic energies of the two"daughter" nuclei is the energy released by the fission of oneuranium-236 nucleus. Calculate the energy released by thefission of 10.0 kg of uranium-236. The atomic mass ofuranium-236 is 236 u, where 1 u = 1 atomic mass unit $$\displaystyle={1.66}\times{10}^{{-{27}}}$$ kg. Express your answer both in joules and in kilotonsof TNT (1 kiloton of TNT releases 4.18 x 10^12 J when itexplodes).
This problem is about the equation
dP/dt = kP-H , P(0) = Po,
where k > 0 and H > 0 are constants.
If H = 0, you have dP/dt = kP , which models expontialgrowth. Think of H as a harvesting term, tending to reducethe rate of growth; then there ought to be a value of H big enoughto prevent exponential growth.
Problem: find acondition on H, involving $$\displaystyle{P}_{{0}}$$ and k, that will prevent solutions from growing exponentially.
$$\begin{array}{|l|l|l|}\hline X&-2&-1&0&1&2\\\hline f(x)&1.125&2.25&4.5&9&18\\\hline g(x)&16&8&4&2&1\\\hline\end{array}$$
Assume that a ball of charged particles has a uniformly distributednegative charge density except for a narrow radial tunnel throughits center, from the surface on one side to the surface on the opposite side. Also assume that we can position a proton any where along the tunnel or outside the ball. Let $$\displaystyle{F}_{{R}}$$ be the magnitude of the electrostatic force on the proton when it islocated at the ball's surface, at radius R. As a multiple ofR, how far from the surface is there a point where the forcemagnitude is 0.44FR if we move the proton(a) away from the ball and (b) into the tunnel?