Module 2: Probabilities and Models

Podcast

Play an AI-generated podcast conversation about this lesson

Questions and Answers

What is the key feature of the exponential distribution that makes it suitable for modeling the time between successive events?

Memorylessness

Given two independent, exponentially distributed random variables A and B with rates α and β, what is the probability that A will win?

α / (α+β)

What is the reliability RX(t) of a component, and what does it represent?

Probability that the component survives till time t

What is the characteristic feature of the exponential distribution in terms of instantaneous failure rate?

Constant instantaneous failure rate λ Signup and view all the answers

What is the distribution of the time between two successive jobs in a computing center, according to the exponential distribution?

Exponential distribution Signup and view all the answers

What is the probability that the sum of two independent exponential random variables A and B is less than or equal to t, given rates α and β?

1 - e^-(α+β)t Signup and view all the answers

What is a model according to the module?

A model is an abstraction of a system that highlights the important features of the system organization and provides ways of quantifying its properties, neglecting all those details that are relevant for the actual implementation, but that are marginal for the objective of the study. Signup and view all the answers

What is a probabilistic model composed of?

A probabilistic model is made up of a list of possible outcomes and the assignment of their respective probabilities. Signup and view all the answers

What is a sample space in a random experiment?

The sample space (S) is the total of the outcomes of the experiment. Signup and view all the answers

What is a series system in terms of component reliability?

A series system works properly if all components work properly. Signup and view all the answers

What is a discrete random variable?

A discrete random variable is a variable in which the sample space image (that is, the set of possible values that it can assume) is finite or numerable. Signup and view all the answers

What is the probability mass function (pmf) of a discrete random variable X?

The pmf, denoted by px, is the Probability that the value of X associated with a result of an experiment is equal to x. Signup and view all the answers

What is the Cumulative Distribution Function (CDF) of a discrete random variable X?

The CDF, denoted by Fx, is the Probability that the value of X associated with an experiment result is less than or equal to x. Signup and view all the answers

What is the unique property of the Geometric distribution?

The Geometric distribution has the particular property called memoryless (loss of memory). Signup and view all the answers

Study Notes

Model and Random Phenomenon

A model is an abstraction of a system that highlights important features and quantifies its properties, neglecting irrelevant details.
Random phenomenon: future behavior is not predictable deterministically.
Probabilistic (or stochastic) model: a list of possible outcomes and their respective probabilities.

Random Experiment and Sample Space

A random experiment is an experiment whose outcome cannot be certain.
A trial is a single execution of the experiment.
The sample space (S) is the total of the outcomes of the experiment.

Component Reliability and System Configurations

Component reliability (Ri) is the probability that component i functions correctly.
Series system: the system works properly if all components work properly.
Parallel system: the system works properly if at least one component works properly.

Random Variables

A random variable X is a function defined on a sampling space S.
A discrete random variable is a variable with a finite or countable sample space image.
Probability mass function (pmf) or probability function (px) is the probability that the value of X is equal to x.
Cumulative Distribution Function (CDF) or Distribution Fx is the probability that the value of X is less than or equal to x.

Geometric Distribution

Geometric distribution: counts the number of trials before the first success in a sequence of Bernoulli trials.
The geometric distribution has the property of memoryless (loss of memory).

Continuous Random Variable and Exponential Distribution

A continuous random variable X is a function that assigns a real number to each element of S.
The exponential distribution is used for modeling events such as time between jobs, service time, time to failure, and repair time.
The exponential distribution is the only continuous distribution with the property of memoryless.
Exponential distribution is characterized by a constant instantaneous failure rate λ.

Independent Exponential Random Variables

Let A and B be two independent exponential random variables, with rates α and β, respectively → 1 - e^-(α+β)t.
Probability that A will win → α / (α+β).

Reliability

Reliability RX(t) is the probability that the component survives till time t.

Studying That Suits You

Use AI to generate personalized quizzes and flashcards to suit your learning preferences.

Description

This quiz covers the basics of probability theory and models, including the definition of a model, its abstraction, and importance in evaluating systems.