Stat 31, Section 1, Last Time

• Big Rules of Probability– The not rule– The or rule– The and rule

P{A & B} = P{A|B}P{B} = P{B|A}P{A}• Bayes Rule

(turn around Conditional Probabilities)• Independence

Independence(Need one more major concept at this level)

An event A does not depend on B, when:

Knowledge of B does not change

chances of A:

P{A | B} = P{A}

New Ball & Urn ExampleH R R R R G G T R R G

Again toss coin, and draw ball:

Same, so R & I are independent events

Not true above, but works here, since proportions of

R & G are same

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Independence

Note, when A in independent of B:

so

And thus

i.e. B is independent of A

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Independence

Note, when A in independent of B:

It follows that: B is independent of A

I.e. “independence” is symmetric in A and B

(as expected)

More formal treatments use symmetric version as definition

(to avoid hassles with 0 probabilities)

Independence

HW:

4.33

Special Case of “And” Rule

For A and B independent:

P{A & B} = P{A | B} P{B} = P{B | A} P{A} =

= P{A} P{B}

i.e. When independent, just multiply probabilities…

Independent “And” Rule

E.g. Toss a coin until the 1st Head appears, find P{3 tosses}:

Model: tosses are independent

(saw this was reasonable last time, using “equally likely sample space ideas)

P{3 tosses} =

When have 3: group with parentheses

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Independent “And” Rule

E.g. Toss a coin until the 1st Head appears, find P{3 tosses}

(by indep:)

I.e. “just multiply”

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Independent “And” Rule

E.g. Toss a coin until the 1st Head appears, P{3 tosses}

• Multiplication idea holds in general

• So from now on will just say:

“Since Independent, multiply probabilities”

• Similarly for Exclusive Or rule,

Will just “add probabilities”

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Independent “And” Rule

HW:

4.31

4.35

Overview of Special Cases

Careful: these can be tricky to keep separate

OR works like adding, for mutually exclusive

AND works like multiplying, for independent

Overview of Special Cases

Caution: special cases are different

Mutually exclusive independent

For A and B mutually exclusive:

P{A | B} = 0 P{A}

Thus not independent

Overview of Special CasesHW: C13 Suppose events A, B, C all have

probability 0.4, A & B are independent, and A & C are mutually exclusive.

(a) Find P{A or B} (0.64)

(b) Find P{A or C} (0.8)

(c) Find P{A and B} (0.16)

(d) Find P{A and C} (0)

Random Variables

Text, Section 4.3 (we are currently jumping)

Idea: take probability to next level

Needed for probability structure of political

polls, etc.

Random Variables

Definition:

A random variable, usually denoted as X,

is a quantity that “takes on values at

random”

Random Variables

Two main types

(that require different mathematical models)

• Discrete, i.e. counting

(so look only at “counting numbers”, 1,2,3,…)

• Continuous, i.e. measuring

(harder math, since need all fractions, etc.)

Random Variables

E.g: X = # for Candidate A in a randomly selected political poll: discrete

(recall all that means)

Power of the random variable idea:

• Gives something to “get a hold of…”

• Similar in spirit to high school algebra:

Give unknowns a name, so can work with

Random Variables

E.g: X = # that comes up, in die rolling:

Discrete

• But not too interesting

• Since can study by simple methods

• As done above

• Don’t really need random variable concept

Random Variables

E.g: X = # that comes up, in die rolling:

Discrete

• But not very interesting

• Since can study by simple methods

• As done above

• Don’t really need random variable concept

Random Variables

E.g: Measurement error:

Let X = measurement:

Continuous

• How to model probabilities???

Random Variables

HW on discrete vs. continuous:

4.40 ((b) discrete, (c) continuous, (d) could be either, but discrete is more common)

And now for something completely different

My idea about “visualization” last time:• 30% really liked it• 70% less enthusiastic…• Depends on mode of thinking

– “Visual thinkers” loved it– But didn’t connect with others

• So don’t plan to continue that…

Random VariablesA die rolling example

(where random variable concept is useful)Win $9 if 5 or 6, Pay $4, if 1, 2 or 3,

otherwise (4) break evenNotes:• Don’t care about number that comes up• Random Variable abstraction allows

focussing on important points• Are you keen to play? (will calculate…)

Random Variables

Die rolling example

Win $9 if 5 or 6, Pay $4, if 1, 2 or 4

Let X = “net winnings”

Note: X takes on values 9, -4 and 0

Probability Structure of X is summarized by:

P{X = 9} = 1/3 P{X = -4} = ½ P{X = 0} = 1/6

(should you want to play?, study later)

Random Variables

Die rolling example, for X = “net winnings”:

Win $9 if 5 or 6, Pay $4, if 1, 2 or 4

Probability Structure of X is summarized by:

P{X = 9} = 1/3 P{X = -4} = ½ P{X = 0} = 1/6

Convenient form: a tableWinning 9 -4 0

Prob. 1/3 1/2 1/6

Summary of Prob. StructureIn general: for discrete X, summarize

“distribution” (i.e. full prob. Structure) by a table:

Where:i. All are between 0 and 1ii. (so get a prob. funct’n as above)

Values x1 x2 … xk

Prob. p1 p2 … pk

11

k

iip

ip

Summary of Prob. Structure

Summarize distribution, for discrete X,

by a table:

Power of this idea:

• Get probs by summing table values

• Special case of disjoint OR rule

Values x1 x2 … xk

Prob. p1 p2 … pk

Summary of Prob. Structure

E.g. Die Rolling game above:

P{X = 9} = 1/3

P{X < 2} = P{X = 0} + P{X = -4} = 1/6 +½ = 2/3

P{X = 5} = 0 (not in table!)

Winning 9 -4 0Prob. 1/3 1/2 1/6

Summary of Prob. Structure

E.g. Die Rolling game above:Winning 9 -4 0Prob. 1/3 1/2 1/6

0

0&90|9

XP

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3

2

2131

31

6131

09

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Summary of Prob. Structure

HW:

4.47 & (d) Find P{X = 3 | X >= 2} (0.24)

4.50 (0.144, …, 0.352)

Random Variables

Now consider continuous random variables

Recall: for measurements (not counting)

Model for continuous random variables:

Calculate probabilities as areas,

under “probability density curve”, f(x)

Continuous Random Variables

Model probabilities for continuous random

variables, as areas under “probability

density curve”, f(x):

= Area( )

a b

(calculus notation)

bXaP

b

a

dxxf )(

Continuous Random Variablese.g. Uniform Distribution

Idea: choose random number from [0,1]

Use constant density: f(x) = C

Models “equally likely”

To choose C, want: Area

1 = P{X in [0,1]} = CSo want C = 1. 0 1

Uniform Random Variable

HW:

4.52 (0.73, 0, 0.73, 0.2, 0.5)

4.54 (1, ½, 1/8)

Continuous Random Variablese.g. Normal Distribution

Idea: Draw at random from a normal population

f(x) is the normal curve (studied above)

Review some earlier concepts:

Normal Curve Mathematics

The “normal density curve” is:

usual “function” of

circle constant = 3.14…

natural number = 2.7…

,2

21

21)(

x

exf

x

Normal Curve Mathematics

Main Ideas:

• Basic shape is:

• “Shifted to mu”:

• “Scaled by sigma”:

• Make Total Area = 1: divide by

• as , but never

2

21x

e

2

0

221 x

e2

21

x

e

0)( xf x

Computation of Normal Areas

EXCEL Computation:

works in terms of “lower areas”

E.g. for

Area < 1.3

)5.0,1(N

Computation of Normal Probs

EXCEL Computation:

probs given by “lower areas”

E.g. for X ~ N(1,0.5)

P{X < 1.3} = 0.73

Normal Random VariablesAs above, compute probabilities as areas,

In EXCEL, use NORMDIST & NORMINV

E.g. above: X ~ N(1,0.5)

P{X < 1.3} =NORMDIST(1.3,1,0.5,TRUE)

= 0.73 (as in pic above)

Normal Random VariablesHW:

4.55