Random Variables

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Random Variables

• Intro to discrete random variables

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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
• What are examples of random variables?
• Let Y ? face showing on die 1,2, , 6

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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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Mean of a Discrete RVInteresting Characteristics

• Expected value of a function of y, a discrete rv
• Let g(y) be function of y

• Suppose C g(T) 5T 3, find E(C)
• E(C) 5(4)30.2 5(5)30.3 5(6)30.5
  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

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Random VariableVariance - A Measure of Dispersion

• Variance of a discrete rv

• Previously defined variance for population
  sample

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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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Deriving the mean and variance of a Bernoulli
Random Variable

• Deriving the mean of a rv

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Deriving the variance of a random variable
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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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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

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Binomial Example Manual solution

• 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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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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Geometric Distribution

• Recall problem statement for geometric
• Suppose p 0.2, what is p(Y3)?
• Only possible order is FFS
• p(Y3) (.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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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(y2)
• Can I use the Binomial distribution???
• Does the p stay constant through all trials??

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Transformer Example

• What do you note about example
• p(defective) changed during sampling process
• of trials n large with respect to N
• What if N gtgt 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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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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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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Class 3 Readings Problems

• Reading assignment
• M S
• Chapter 4 Sections 4.1 - 4.10
• Recommended problems
• M S Chapter 4
• 59, 69, 84, 87, 88, 90, 96, 98, 100