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R2. Risk in Elementary Events Decisions, Uncertainty, Anticipation, Sampling. 1/7. Hazard….vs…Realization of Hazard. Idea of Chance. Kinds of Problems to be Addressed: (a) Expose a population to a new drug with a side-effect that can be fatal What fraction of those exposed will die?

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R2

Risk in Elementary Events

Decisions, Uncertainty, Anticipation, Sampling

1/7

Hazard….vs…Realization of Hazard

Idea of Chance

• Kinds of Problems to be Addressed:
• (a) Expose a population to a new drug with a side-effect that can be fatal
• What fraction of those exposed will die?
• If I take this drug what is the chance I will die?
• How do I interpret this chance to decide whether to take it or not?

(b) As a California resident I am exposed to earthquake hazards

• What is the chance I will experience an R’>9 in my lifetime?
• How do I interpret this chance in order to decide if I want to stay or leave?
• What is the chance of an R > 9 tomorrow? Within next year? Next 10 yrs?

Binary Answer: {S, P, C} and {S, , } P + =1

In common language this type of Risk is referred to simply by p.

Population Sampling or an Event Occurring over Time

R2

Risk in Elementary Events

Decisions, Uncertainty, Anticipation, Sampling

Examples

1a/7

• Kinds of Problems to be Addressed:
• Expose a population to a new drug with a side-effect that can be fatal
• p= 10-5 the Risk is 1 in 100,000 that I will die
• Thinking of my personal risk….if I had 100,000 life-times to live….
• Am I of above or below “normal” resistance? How confident am I of the 10-5 ?
• What would happen if I did not take the drug?
• (b)As a California resident I am exposed to earthquake hazards
• l= 10-2per year found to be an average rate….once in a 100 years
• If I live 30 more years my chance is 30/100 or ~30%....I would expect
• the event on the average once every 3 life-times of 30 years each.
• What about tomorrow? Next year? Next 10 years?
• How confident am I to use the 10-2 for such predictions? The model?
• What a possible realization of the next 30 years might look like?

R2

Risk in Elementary Events

Experiments, Experience, Judgment, Sampling, FPS

2/7

• The Dual Use of Probability
• To express results of Sampling from a Fundamental Probability Set (FPS)
• It is an empirically-driven endeavor carried out mostly on statistical basis
• A FPS can be thought of as a real, binary, stable population. A random
• sampling from it is subject to the laws of probability theory, and it is
• predictable.
• Such a population can be characterized to any degree of precision by
• random sampling, and statistical inference. Then I can predict.
• Aleatory Uncertainty
• (b) To express “Expert Opinion” about the chance of a future Event.
• Experts use Theory, Numerical and Experimental Simulations, and Judgment
• The involves a fictitious FPS, a theoretical construct, that allows us to treat real, and very complex events as if they were stable, and subject to some statistical law that can be inferred from the past experience.
• Epistemic Uncertainty

R2

Risk in Elementary Events

The Fundamental Probability Set

3/7

Take a random sample of size N from the population

Find f(b) =

Define p(b) =

Bernoulli’s First Limit Theorem

Frequency: can be used to estimate p

Probability: can be used to define chance in future draws

Think of as an event

f is Empirical p is Theoretical

R2

Risk in Elementary Events

Inference

3a/7

Observed frequency is

taken as a surrogate

for p

Fisherian

or

Classical

Approach

Sampling

f

A real

FPS…p

Confidence Intervals

Aleatory Uncertainty

Sampling

+ Other considerations

Estimated “p” is taken directly as p

“p”

An assumed

FPS….””p””

Bayesian

Approach

Bayes Theorem

Epistemic Uncertainty

R2

Risk in Elementary Events

Inference, Prediction, and Simulation (Realizations)

3b/7

• We will focus on three kinds of Random Processes
• Bernoulli Process …..Binomial Distribution…it is discrete
• Poisson Process…….Poisson Distribution…discrete
• Underlying a Poisson Process….Exponential Distribution…Continuous
• Gaussian Process…..Normal Distribution…..continuous
• Uniform Randomness….Uniform Distribution….continuous or discrete
• These will serve as Models to use for Prediction
• For this we will need
• to Select an appropriate Model and
• to Determine the Parameters appropriate to the application
• The latter part involves Sampling and Inference.
• This in turn involves the concept and use of Confidence Level

R2

Risk in Elementary Events

Bernoulli Process. Binomial Distribution.

2/7

What if N is finite and small?

If population is p, what is P for f=0? 1?

Any other 0<f<1?

We take random samples of size N, over and over again and

we look for fraction of times the event

{# of black equal to Nb } has occurred. This is equal to

Nomenclature: p, P (prob. Of an Event), p’ (prob. Density)

t

R2

Risk in Elementary Events

Poisson Process and Distribution.

4/7

T

Events of interest are occurring randomly in time

but independently of each other and at a steady average rate of levents per unit time.

Where r is the number of events in time T, and

l is the Rate or Frequency of the Poisson Process

R2

Risk in Elementary Events

Exponential Distribution

4a/7

T

Events are occurring randomly on a time axis and

according to a Poisson process.

T is the inter-event time interval—the random variable

T follows the Exponential Distribution.

There is no memory. To find probability of the event occurring

in time T integrate the density function from 0 to T.

R2

Risk in Elementary Events

Gaussian Process and Normal Distribution

5/7

A Gaussian Process can be thought of as the cumulative

result of many random influences of equal magnitude but

some positive and some negative…like molecular diffusion!

The density function is called Nornal.

Where m ands are the mean and standard variation.

The “standard error” (Z) at position x describes the distance

from the mean measured in terms of s.

The following few slides are to

illustrate shapes of distributions

of special interest to us.

Note that MATLAB calls the

Binomial and Poisson functions

“pdf’s”. This is for convenience.

They are NOT pdf’s.

They are discrete distributions.