random variables

statistical processes 1

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

What are examples of random variables?

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


VariablesA Simple Taxonomy

Variables

Random Variables Deterministic variables

Discrete random variables

Continuousrandom variables

Variables arebut models

Variables arebut models


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?


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!

Note conventions!


Characteristic Measures for PMFsCentral Tendency

Central tendency of a pmf

Mean or average

y all

)]([)( ypyyE

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)?


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)+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

y all

)]()([)]([ ypygygE


Random VariableVariance - A Measure of Dispersion

Variance of a discrete rv

Previously defined variance for population & sample

jy all

2j

22 )()-(y)( jypyE

1

)(

:Variance Sample

)()(

:Variance Population

1

2

2

1

2

1

2

2

n

yys

n

d

n

y

n

i

i

n

i

i

n

i

i


Mean and VarianceInterpretation

Mean

Expected value of the random variable

Variance

Expected value of distance2 from mean

22 )(

)(

yE

yE


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


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


Probability Mass FunctionBernoulli RV

2)(

)(

1 y for p

0 yfor q)(

occur doesevent if S,or 1

occurnot doesevent if F,or 0

pqyVAR

pyE

yP

y

yHow can we derive these?


Deriving the mean and variance of a Bernoulli Random Variable

Deriving the mean of a rv:

ppppyypyE

yypyE

qp

ppyp

y

y

)1()0(0)1(1)()(

)()(

)0(

)1()1(

We know that

and

We also know that

So it follows that


Deriving the variance of a random variable

pqpppp

p

pp

ypyyE

)1(

)1(

)0()0()1()1(

)()(

22

22

2222

22222


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


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

3!1!2

!3

)!(!

!

2

3

nNn

N

Returning to the P(2 heads)

375.0)5.0()5.0(32

3 heads) P(2 212

qp


Binomial Distribution FunctionCreating the Model

Key assumption

Each trial an independent, identical Bernoulli variable

E(y) = np

Var(y) = npq

ynyqpy

n

P(y)

trialsBernoullin in successes of # y Let

y n

# of combinationsProbability of y successes

Probability of n - y failures


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


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!

176.0)756,184(

756,184!10!10

!20

10

20

P(y)

1010

qp

y

n

qpy

n yny


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?


Formula for MultinomialUnderstand Relationship to Binomial

kyk

yy

k

k

k

pppyyy

n

yyyp

yYyYyYp

...!,...,!,!

!

),...,,(

),...,,(

2121

21

21

21

This is calleda joint distribution.

Need to understand conventionNote there are k random variables

j = npj j2 = npjqj = npj(1-pj)


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


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?


Negative Binomial Distribution

2

1

1)(

p

rq

p

r

qpr

yyp ryr

ProblemHave series of Bernoulli trials, want probability of waiting until yth trial to get rth success

Let p = 0.5 & r = 2, do we getreasonable results?


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??


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


Hypergeometric Distribution

n

N

y

r

ypynrN

)(

N # in populationn # in sampler # of Successes in populationy # of Successes in sample

1) Why is y a rv?2) What do we mean byp(y)?3) What is r/N ?


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?


A Poisson Process is a Rate

# of cars passing a fixed point in one

minute

# of defects in an 8x8 sheet of plywood


Poisson Probability Distribution

2

!)(

y

eyp

y

Where,y # of occurrences in a given unit mean # of occurrences in a given unite 2.71828…

Note particularlyinteresting relationship

Note must be for the same unit of measure!

Why does thismake sense?


Discrete Random VariablesExcel Special Functions

SpecialFunctions

SpecialFunctionsExcel

HYPGEOMDISTBINOMDISTNEGBINOMDISTPOISSON

Are there others?


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

random variables

Documents

function of y

random variablesintro

y face

examples of random variables

probability of y successes

linear operatorex y

discrete rvpreviously

math function