Answered: 3. The storm runoff X (in cubic meters…

3. The storm runoff X (in cubic meters per second, ems) can be modeled by a random variable with the following probability density function (PDF): fx(x) = c (x -) for 0

A First Course in Probability (10th Edition)

10th Edition

ISBN:9780134753119

Author:Sheldon Ross

Publisher:Sheldon Ross

Chapter1: Combinatorial Analysis

Section: Chapter Questions

Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...

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Concept explainers

Continuous Probability Distributions

Probability distributions are of two types, which are continuous probability distributions and discrete probability distributions. A continuous probability distribution contains an infinite number of values. For example, if time is infinite: you could count from 0 to a trillion seconds, billion seconds, so on indefinitely. A discrete probability distribution consists of only a countable set of possible values.

Normal Distribution

Suppose we had to design a bathroom weighing scale, how would we decide what should be the range of the weighing machine? Would we take the highest recorded human weight in history and use that as the upper limit for our weighing scale? This may not be a great idea as the sensitivity of the scale would get reduced if the range is too large. At the same time, if we keep the upper limit too low, it may not be usable for a large percentage of the population!

Question

3. The storm runoff X (in cubie meters per second, ems) can be modeled by a random variable with the following
probability density function (PDF):
fa(x) = c (x -)
for 0 <x< 8
= 0
otherwise
a) Determine the constant e.
b) Sketch the PDF (bonus).
c) Find the mean, variance and median of X.
d) The pipe carrying the runoff has a capacity of 5 cms. What is the probability that overflow occurs after a storm
(the pipe capacity being inadequate)?
e) If overflow occurs after a storm, what is the probability that the runoff from a storm is less than 7 cms?
f) An engineer considers replacing the current pipe by a larger pipe having a capacity of 6 cms. Suppose there is
a probability of 60% that the replacement would be completed prior to the next storm. What is the probability
of overflow in the next storm?

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