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

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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?
Transcribed Image Text: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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