Aperçu des semaines
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The use of probability models and statistical methods for analyzing data has become
common practice in virtually all scientific disciplines. This course attempts to provide
a comprehensive introduction to those models and methods most likely to be encoun-
tered and used by students in their careers in engineering and the natural sciences.
Although the examples and exercises have been designed with scientists and engi-
neers in mind, most of the methods covered are basic to statistical analyses in many
other disciplines, so that students of business and the social sciences will also profit
from reading the book.
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Introduction to probability and basic definitions
- Random Experiments:
- An experiment that can result in different outcomes, even though it is repeated in the same manner every
time, is called random experiment.
-Sample Space:
- The set of all possible outcomes of a random experiment is called the sample space of the experiment. The
sample space is denoted S.
- A sample space is said discrete if it consists of a finite or countable infinite set of outcomes.
- A sample space is said continuous if it contains an
interval (either finite or infinite) of a real numbers.
Examples:
1. Tossing a coin
2. Tossing a die3. Response of a patient to a treatment (cure, no cure).4. Presence of a genetic trait in a newborn baby.
Sample space (S)
1. Tossing a coin :
2. Tossing a die :
S={H, T}
S = {1,2,3,4,5,6}
3. Response of a patient to a treatment :
S = {cure, no cure}
1-Consider an experiment in which a process
Chapter 2 : Probability
II
Définition : Sample spaces and Events
Exemple : Basic examples
Exemple : Example
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manufacturers pins whose lengths vary between
5.20 mm and 5.25 mm. The sample space as simply the positive real line
S={x|5.20<x<5.25}
2- If the objective of the analysis is to consider only whether a particular still pin too short, too long or within
specifications, the sample space might be taken to be the set of three outcomes:
S={too short, too long, within specification}
1. Sample spaces can also be described graphically with tree diagrams.
2. When a sample space can be constructed in several steps or stages, we can represent each of the n1 ways of
completing the first step as a branch of a tree.
3. Each of the ways of completing the second step can be represented as n2 branches starting from the ends of
the original branches, and so forth.
1.1. Events
1.An event is a subset of the sample space of a random experiment.
Example:
Consider the sample space S = {yy, yn, ny, nn}.
2. Suppose that the set of all outcomes for which at least one part conforms is denoted as E1.
E1={yy, yn, ny}.
3. The event in which both parts do not conform,
denoted as E2, contains only the single outcome,
E2={nn}.
4. Other examples of events are E3=o, the null set,
and E4=S, the sample space.
1.2. Operations and Events
1. The union of two events is the event that consists of all outcomes that contained in either of the two events.
We denote the union as
𝐴 u 𝐵
2. The intersection of two events is the event that
consists of all outcomes that are contained in both of the two events. We denote the intersection as
𝐴 u 𝐵
3. The complement of an event in a sample space is the set of outcomes in the sample space that are not in the
event. We denote the complement of the event 𝐴 and 𝐴𝑐.
Tree Diagrams
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1.3. Interpretations of Probability
1. Probability refers to the chance that a particular
event will occur.
2. Let A be an event, then P(A) denotes the probability that A will occur.
3.Used to quantify likelihood or chance.
4. Used to present risk or uncertainty in engineering applications.
5.Can be interpreted as our degree of belief or relative frequency.
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In this chapter, we provide a general review of probability techniques.