Since \[F_\mathcal{D}(x) = \int_{-\infty}^x f_\mathcal{D}(u)du,\]
if \(f_\mathcal{D}\) is continuous at \(x\), the Fundamental Theorem of Calculus gives: \[f_\mathcal{D}(x) = \frac{d}{dx}F_\mathcal{D}(x).\]
Quantiles
The quantile function is the inverse of the CDF:
\[q(\alpha) = F^{-1}_\mathcal{D}(\alpha)\]
So \[x_0 = q(\alpha) \iff \mathbb{P}_\mathcal{D}(X < x_0) = \alpha.\]
(a) Relationship of CDF and PDF
(b)
Figure 2
Distributions Are Assumptions
Specifying a distribution is making an assumption about observations and any applicable constraints.
Examples: If your observations are…
Continuous and fat-tailed? Cauchy distribution
Continuous and bounded? Beta distribution
Sums of positive random variables? Gamma or Normal distribution.
Selecting a Distribution
A distribution implicitly answers questions like:
What is the most probable event? How much more likely is it than the others?
Are larger or smaller events more, less, or equally probable?
How probable are extreme events?
Are different events correlated, or are they independent?
Probability Distribution Tails
The tails of distributions represent the probability of high-impact outcomes.
Key consideration: Small changes to these (low) probabilities can greatly influence risk.
Figure 3: Comparison of Cauchy and Normal tails
“What Distribution Should I Use?”
There is no right answer to this, no matter what a statistical test tells you.
What assumptions are justifiable?
Particularly: think about tails, symmetry/skew
What information do you have?
“What Distribution Should I Use?”
For example, suppose our data are counts of events:
If you know something about rates, you can use a Poisson distribution
If you know something about probabilities, you can use a Binomial distribution.
Conditional Probabilities
We often don’t want to just know if a particular event \(A\) has a certain probability, but also how other events (call them \(B\)) might depend on that outcome.
In other words:
We want the conditional probability of \(B\) given \(A\), denoted \(\mathbb{P}(B|A)\).
Conditional Probabilities
We can write conditional probabilities in terms of unconditional probabilities:
Finding \(\pi\) by sampling random values from the unit square and computing the fraction in the unit circle. This is an example of Monte Carlo integration.
\[\frac{\text{Area of Circle}}{\text{Area of Square}} = \frac{\pi}{4}\]
Figure 4: Monte Carlo simualtion for Estimating pi
MC Example: Dice
What is the probability of rolling 4 dice for a total of 19?
Can simulate dice rolls and find the frequency of 19s among the samples.
Figure 5: Monte Carlo simualtion for Dice Rolls
MC Estimate of Quantiles
Would like to estimate the CDF \(F\) with some approximation \(\hat{F}_n\), then compute \(\hat{z}^\alpha_n = \hat{F}_n^{-1}(\alpha)\) as an estimator of the \(\alpha\)-quantile \(z^\alpha\).
This type of estimation can be repeated with any simulation model that has a stochastic component.
For example, consider our dissolved oxygen model. Suppose that we have a probability distribution for the inflow DO.
How could we compute the probability of DO falling below the regulatory standard somewhere downstream?
Monte Carlo and Uncertainty Propagation
Draw samples from some distribution;
Run them through one or more models;
Compute the (conditional) probability of outcomes of interest (for good or bad).
Key Takeaways
Key Takeaways
Systems analysis features many uncertainties, which may be neglected with certain methods/model setups.
Risk: Combination of hazard, exposure, vulnerability, and response.
Choice of probability distribution can have large impacts on uncertainty and risk estimates: try not to use distributions just because they’re convenient.
Monte Carlo: Estimate expected values of outcomes using simulation.
Upcoming Schedule
Next Classes
Monday: Monte Carlo Lab (clone before class, maybe instantiate environment too)
