[music playing]
In the previous class, we learned about Probability
Density Functions, or PDFs.
We looked at various types of PDFs,
including binomial distributions,
normal distributions, and lognormal distributions.
And we saw that we could model the probability
of a particular outcome such as the number of tails
results in five coin flips as a PDF relating
the range of various possible outcomes
to the expected frequency of occurrence.
In many real world problems, we may
need to consider multiple factors that can
be best represented by a PDF.
Imagine I wanted to determine the likelihood of Sydney's
main water supply, Warragamba Dam, overflowing
and leading to flooding in downstream places
like Penrith, Richmond, or Windsor.
The likelihood of the dam overflowing
at any particular time depends on a number of factors.
One factor is how much rain falls
in the catchment over some specified period of time.
We could also think about how dry the catchment is
before the rainfall which will affect how much runoff
is generated to actually flow into Warragamba Dam,
but we also need to consider how full the dam is
at any particular time and therefore
how much available capacity it has
to hold back the flood waters.
All of these factors vary over time,
so their state at any particular moment
is largely a matter of chance.
This chart shows how much water has been in the dam
over a 20-year period.
It also shows the full operating storage as the green line.
We could use this information to produce a PDF for the reservoir
volume available to capture inflows
at any particular moment in time.
We can see that such a PDF would range from 0 megaliters
to around 1 and 1/2 million megaliters.
It might look something like this.
Now let's say we have a 24-hour rainfall event that
leads to a certain volume of runoff
being generated and flowing into the dam.
Rainfall over 24 hours is highly variable,
so we could represent that by a PDF which
might look something like this.
How can I determine the likelihood
of too much water flowing into the dam,
exceeding the available capacity, and the dam spilling?
To calculate the overflow volume, if any,
I could use the following simple equation.
The overflow volume in megaliters
equals the inflow volume minus the available storage capacity.
But what numbers should I plug into this formula?
At any particular time, both the inflow volume
and the available storage capacity
could be any value on their respective PDFs.
If at a particular time runoff is low and available
storage capacity is high, the dam won't spill.
If storage capacity is low and runoff is high, it will.
But what are the chances?
With what frequency should I expect spill conditions
to occur?
To answer these questions, I need
to develop a PDF showing the calculated overflow volume.
In this case, the two parameters in the equation are also PDFs.
I need to subtract the available storage capacity
PDF from the inflow volume PDF.
Can you do that mathematically?
What if the PDFs are completely different shapes?
How would you do it?
In this class, I'd like you to come up
with one example of your own where
a final PDF needs to be derived from two or more input PDFs.
What is it that you want to produce a PDF from,
and from what type of PDF data could you produce it?
Can you give the equation that you would use like the one
that I gave to derive dam overflow volume?
Any examples would be acceptable.
They don't have to be based on engineering concepts.
You can think of an example if you try.
As usual, I'll ask a few people to present their examples
at the start of the class.