Ch3_RandomVariables_ProbIntro_torresgarcia.pdf

7/24/2019 Ch3_RandomVariables_ProbIntro_torresgarcia.pdf

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Reference: Most slides adapted from Montgomery et al. (2011)

CHAPTER 3:RANDOM VARIABLES AND

PROBABILITY INTRO

Instructor:

Wandaliz Torres-García, Ph. D.


ININ 5559ENGINEERING STATISTICS




RANDOM VARIABLES AND PROBABILITYDISTRIBUTIONS1. Random Numbers, Random Variables (RVs), Random Experiment (3-1 & 3-2)

2. Probability Laws (3-3)

3. Continuous RVs (3-4 & 3-5) CRVs Distributions: Normal, Lognormal, Gamma, Weibull, Beta

4. Probability Plots (3-6)

5. Discrete RVs (3-7 & 3-8) DRVs Distributions: Binomial

6. Poisson Process (3-9)

7. Normal Approximation to the Binomial and Poisson Distributions (3-10)

8. More than one RV and Independence (3-11)

9. Functions of RVs (3-12)

10. Statistics and the Central Limit Theorem (3-13)



3-1 Introduction




WHAT IS A RANDOM VARIABLE?

Random Variable: A variable that assumes different values withspecific probabilities.

In a random experiment, a variable whose measured value canchange (from one replicate of the experiment to another) is referredto as a random variable.

There are two types of random variables, discrete and continuous.

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DISCRETE VS. CONTINUOUSDiscrete Random Variable: A random variable whose values arefinite or countable infinite.

Finite values example:

Academic status (freshman, sophomore, junior, senior, grad)

Number of children born to a couple Countable infinite values example:

Number of people who have been born since mankind began

Set of integers

Continuous Random Variable: A random variable that can assume

values on an interval (finite or infinite). Interval values example:

Waiting time at the ATM

Production Yield (# of items passing inspection/# items tested)

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http://slidepdf.com/reader/full/ch3randomvariablesprobintrotorresgarciapdf 7/26 Reference: Most slides adapted from Montgomery et al. (2011)

DISCRETE OR CONTINUOUS FUNCTIONS

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0for x2)( 2

xe x f

6...,2,1,zfor61)( z p

..321 )1()( 1 , . , , for x p p x p x

164 12

1)( y for y f

Discrete

Discrete

Continuous

Continuous


http://slidepdf.com/reader/full/ch3randomvariablesprobintrotorresgarciapdf 8/26 Reference: Most slides adapted from Montgomery et al. (2011)

MORE EXAMPLES

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DISCRETE

DISCRETE

CONTINUOUS

CONTINUOUS




3-3 Probability

•

Used to quantify likelihood or chance• Used to represent risk or uncertainty in engineering

• Can be interpreted as our degree of belief or relative frequency

• Probability statements describe the likelihood that particular

values occur.

• The likelihood is quantified by assigning a number from the

interval [0, 1] to the set of values.

• Higher numbers indicate that the set of values is more likely.




3-3 Probability




3-3 Probability

An Event is a set of outcomes from an experiment or a subset of the

sample space.

Complement of an Event

• Given a setE , the complement of

Eis the set of elements that are not

in E . The complement is denoted as E ’.

Mutually Exclusive Events

• The sets E 1 , E 2 ,...,E k are mutually exclusive if the intersection ofany pair is empty. That is, each element is in one and only one of the

sets E 1 , E 2,...,E k .




3-3 Probability

PROBABILITY PROPERTIES




EXAMPLES:




RANDOM VARIABLES AND PROBABILITYDISTRIBUTIONS1. Random Numbers, Random Variables (RVs), Random Experiment (3-1 & 3-2)

2. Probability Laws (3-3)

3. Continuous RVs (3-4 & 3-5) CRVs Distributions: Normal, Lognormal, Gamma, Weibull, Beta

4. Probability Plots (3-6)

5. Discrete RVs (3-7 & 3-8) DRVs Distributions: Binomial

6. Poisson Process (3-9)

7. Normal Approximation to the Binomial and Poisson Distributions (3-10)

8. More than one RV and Independence (3-11)

9. Functions of RVs (3-12)

10. Statistics and the Central Limit Theorem (3-13)



Reference: Most slides adapted from Montgomery et al. (2011) 15




IN GENERAL

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P(a < X < b) is the same value as

P(a < X < b) is the same value as

P(a < X < b) is the same value asP(a < X < b)

bx

ax

f(x)dx

All are computed this way

0.1)( xall

dx x f




3-4 Continuous Random Variables

3-4.1 Probability Density Function





EXAMPLE 3-3: FLAW ON MAGNETIC DISK




EXAMPLES: CONTINUOUS RANDOMVARIABLES AND THEIR PDFS

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X ~ Exponential Distribution

X ~ Uniform Distribution

X ~ Normal Distribution

“ X is distributed normally”

“ X is distributed uniformly”

“ X is distributed exponentially”




CUMULATIVE DISTRIBUTION FUNCTION (CDF)

OF CONTINUOUS RANDOM VARIABLES

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In practical use, this

is really the lowest x value.

These properties hold for BOTH discrete




PROPERTIES OF CDF’S

0 < F(x) < 1 for - < x < +

Translation: F(x) is a probability. It is the probability that random

variable X (upper case) is < a specific value, x (lower case). Since F(x)

is a probability, it must lie between 0 and 1.

limx + F(x) = 1

Translation: As X approaches its largest value (which might or might notbe infinity), F(x) approaches 1.0. When X is at its largest value or

beyond, F(x) = 1.

limx - F(x) = 0

Translation: As X approaches its smallest value (which might or might not

be negative infinity), F(x) approaches 0. When X is at its smallest valueor below, F(x) = 0.

F(x) is non-decreasing

Translation: As we move to the right, increasing the value of X, if x2 >

x1, then F(x2 ) > F(x1 )

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These properties hold for BOTH discrete

and continuous random variables




EXAMPLE 3-3: FLAW ON MAGNETIC DISK





EXAMPLE 3-3





3-4.3 MEAN AND VARIANCE



Reference: Most slides adapted from Montgomery et al (2011)

WHAT IS THE SIGNIFICANCE OF EXPECTED

VALUE AND OF VARIANCE?The expected value is a weighted average of all the possible values a

random variable, (discrete or continuous), can assume.The expected value is often used as a measure of a modeled system, e.g., mean number of

defects, mean waiting time in a queue.

The variance is a measure of the spread of the data and is used as a

measure to characterize a distribution or a modeled system.

For example, a random variable with high variance indicates that the values

can take on a wide range of possible values.

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