Random variables
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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

Random Variables

Intro to discrete random variables

statistical processes


Random variables1

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}

statistical processes


Variables a simple taxonomy

Random Variables

Deterministic variables

Continuous

random variables

Discrete

random variables

VariablesA Simple Taxonomy

Variables are

but models

Variables

statistical processes


Random variables a simple example

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?

statistical processes


Developing rv standard models distribution functions

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 pmfs central tendency

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)?

statistical processes


Mean of a discrete rv interesting 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)+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

statistical processes


Random variable variance a measure of dispersion

Variance of a discrete rv

Previously defined variance for population & sample

Random VariableVariance - A Measure of Dispersion

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Mean and variance interpretation

Mean and VarianceInterpretation

  • Mean

    • Expected value of the random variable

  • Variance

    • Expected value of distance2 from mean

statistical processes


Discrete random variables useful models

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

statistical processes


Binomial distribution function setting the stage

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 function bernoulli rv

    How can we derive these?

    Probability Mass FunctionBernoulli RV

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    Deriving the mean and variance of a bernoulli random variable

    We know that

    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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    Deriving the variance of a random variable

    Deriving the variance of a random variable

    statistical processes


    Binomial distribution problem description

    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

    statistical processes


    Binomial distribution function generalizing from simple example

    Returning to the P(2 heads)

    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

    statistical processes


    Binomial distribution function creating the model

    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

    statistical processes


    Binomial distribution function simple problem

    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


    Binomial example manual solution

    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!

    statistical processes


    Random variables

    statistical processes


    Random variables

    statistical processes


    Random variables

    statistical processes


    Multinomial distribution generalizing the binomial distribution

    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?

    statistical processes


    Formula for multinomial understand relationship to binomial

    Need to understand convention

    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)

    statistical processes


    Extending the binomial two special cases

    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

    statistical processes


    Geometric distribution

    Geometric Distribution

    • 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?

    statistical processes


    Negative binomial distribution

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    reasonable results?

    Negative Binomial Distribution

    Problem

    Have series of Bernoulli trials, want probability of waiting

    until yth trial to get rth success

    statistical processes


    Hypergeometric an extension to the binomial

    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??

    statistical processes


    Transformer example

    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 >> 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

    statistical processes


    Hypergeometric distribution

    1) Why is y a rv?

    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

    statistical processes


    Poisson process a useful model

    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 defects in an 8x8 sheet of plywood

    A Poisson Process is a Rate

    # of cars passing a fixed point in one minute

    statistical processes


    Poisson probability distribution

    Why does this

    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…

    statistical processes


    Discrete random variables excel special functions

    Discrete Random VariablesExcel Special Functions

    Special

    Functions

    HYPGEOMDIST

    BINOMDIST

    NEGBINOMDIST

    POISSON

    Are there others?

    Excel

    statistical processes


    Class 3 readings problems

    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

    statistical processes


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