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# Random Variables - PowerPoint PPT Presentation

Random Variables. Intro to discrete random variables. Random Variables. “A random variable is a numerical valued function defined over a sample space” What does this mean in English? If Y  rv then Y takes on more than 1 numerical value Sample space is set of possible values of Y

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### Random Variables

Intro to discrete random variables

statistical processes

• “A random variable is a numerical valued function defined over a sample space”

• What does this mean in English?

• If Y  rv then Y takes on more than 1 numerical value

• Sample space is set of possible values of Y

• What are examples of random variables?

• Let Y  face showing on die ={1,2, …, 6}

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Deterministic variables

Continuous

random variables

Discrete

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VariablesA Simple Taxonomy

Variables are

but models

Variables

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Random VariablesA Simple Example

• Variables model physical processes

• Let S  sales; C  costs; P  profit

P = S - C

• Suppose all variables deterministic

• S = 25 and C = 15,  P = 10

• Suppose S is a rv = {25, 30}

• What is P?

• RVs may be used just as deterministic variables

• How shall we describe the behavior of a rv?

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Developing RV Standard ModelsDistribution Functions

• Distribution functions assign probability to every real numbered value of a rv

• Probability Mass Function (PMF) assigns probability to each value of a discrete rv

• Probability Density Function (PDF) is a math function that describes distribution for a continuous rv

• Standard models convenient for describing physical processes

• Example of PMF: Let T  project duration (a rv)

• t1 = 4 weeks; p(T = t1) = p(t1) = 0.2

• t2 = 5 weeks; p(T = t2) = p(t2) = 0.3

• t3 = 6 weeks; p(T = t3) = p(t3) = 0.5

Note

conventions!

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Characteristic Measures for PMFsCentral Tendency

• Central tendency of a pmf

• Mean or average

• What is E(T) for project duration example?

•  = 4(0.2) + 5 (0.3) + 6(0.5) = 5.3 weeks

• What if C = f(T), where C  costs

• Is C a random variable?

• What is E(C)?

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Let g(y) be function of y

Suppose C = g(T) = 5T + 3, find E(C)

E(C) = [5(4)+3]0.2 + [5(5)+3]0.3 + [5(6)+3]0.5 = 29.5

Let d = constant

E(d)= constant

E(dy)= dE(y)

E() is a linear operator

E(X + Y) = E(X) + E(Y), where X & Y are rv

Mean of a Discrete RVInteresting Characteristics

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Previously defined variance for population & sample

Random VariableVariance - A Measure of Dispersion

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Mean and VarianceInterpretation

• Mean

• Expected value of the random variable

• Variance

• Expected value of distance2 from mean

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Discrete Random VariablesUseful Models

• Examine frequently encountered models

• Be sure to understand

• Process being modeled by random variable

• Derivation of pmf

• Use of Excel

• Calculating pmf

• Graphing pmf

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Binomial Distribution FunctionSetting the Stage

• Bernoulli rv

• Models process in which an outcome either happens or does not

• A binary outcome

• What are examples?

• Formal description

• Trial results in 1 of 2 mutually exclusive outcomes

• Outcomes are exhaustive

• P(S) = p ; P(F) = q ; p + q = 1.0

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Probability Mass FunctionBernoulli RV

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and

We also know that

So it follows that

Deriving the mean and variance of a Bernoulli Random Variable

• Deriving the mean of a rv:

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Binomial DistributionProblem Description

Problem:

• Given n trials of a Bernoulli rv, what is probability of y successes?

• Why is y a discrete rv?

• Simple example

Toss coin 3 times, find P(2 heads)

n = 3 ; y = 2

P(H, H, T) = (.5)(.5)(.5) = 0.125

Could also be (H,T,H) or (T, H, H)

P(2 heads) = 0.125 + 0.125 + 0.125 = 0.375

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Binomial Distribution FunctionGeneralizing From Simple Example

• Recall 2 heads in three tosses

• How many different ways is this possible?

• Combination of three things taken two at a time

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y  n

Probability of y successes

# of combinations

Probability of n - y failures

Binomial Distribution FunctionCreating the Model

• Key assumption

• Each trial an independent, identical Bernoulli variable

• E(y) = np

• Var(y) = npq

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Binomial Distribution FunctionSimple Problem

Have 20 coin tosses

• Find probability that will have 10 or more heads

• Set up the problem and will then solve

• Let

• n = 20

• y = # of heads

• p = q = 0.50

• Want p(y  10)

• Will solve manually and using Excel

• statistical processes

• But remember! This is just for y = 10. We must do this for y = 11, 12, …, 20 as well and then sum all the values!

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Multinomial DistributionGeneralizing the Binomial Distribution

Problem

Events E1, …, Ek occur with probabilities p1, p2, …, pk . Given n independent trials probability E1 occurs y1 times, … Ek occurs yk times.

• Why is this a more general case than the Binomial?

• Can you describe an example?

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Note there are k random variables

This is called

a joint distribution.

Formula for MultinomialUnderstand Relationship to Binomial

j = npj j2 = npjqj = npj(1-pj)

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Extending the BinomialTwo Special Cases

• Recall Binomial distribution

• What problem does it model?

• Given n independent trials, p = p(success)

• Geometric distribution

• Define y as rv representing first success

• Negative Binomial

• Define y as rv representing rth success

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• Recall problem statement for geometric

• Suppose p = 0.2, what is p(Y=3)?

• Only possible order is FFS

• p(Y=3) = (.8)(.8)(.2)

• Generalizing simple example

• p(y) = pqy-1 ;  = 1/p ; 2 = q / (p2)

• What is implicit assumption about largest value of y?

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Let p = 0.5 & r = 2, do we get

reasonable results?

Negative Binomial Distribution

Problem

Have series of Bernoulli trials, want probability of waiting

until yth trial to get rth success

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HypergeometricAn Extension to the Binomial

• Suppose have 10 transformers, know 1 is defective

• p(defective) = 0.1

• Let y = # of defectives in a sample of n

• Suppose pick 3 transformers, find p(y=2)

• Can I use the Binomial distribution???

• Does the p stay constant through all trials??

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• What do you note about example:

• p(defective) changed during sampling process

• # of trials n large with respect to N

• What if N >> n ?

• Would p(defective) change during sampling process?

• Process called sampling without replacement

• Binomial assumes infinite population OR sampling with replacement. Why?

• If we cannot use Binomial then what?

Hypergeometric Probability Distribution

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2) What do we mean by

p(y)?

3) What is r/N ?

Hypergeometric Distribution

N  # in population

n  # in sample

r  # of Successes in population

y  # of Successes in sample

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Poisson ProcessA Useful Model

• In a Poisson process

• Events occur purely randomly

• Over long term rate is constant

• What is implication of the above?

• Memoryless process

• What are some processes modeled as Poisson processes?

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A Poisson Process is a Rate

# of cars passing a fixed point in one minute

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make sense?

Note particularly

interesting relationship

Note must be for

the same unit of

measure!

Poisson Probability Distribution

Where,

y  # of occurrences in a given unit

  mean # of occurrences in a given unit

e  2.71828…

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Discrete Random VariablesExcel Special Functions

Special

Functions

HYPGEOMDIST

BINOMDIST

NEGBINOMDIST

POISSON

Are there others?

Excel

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