last time administrative matters – blackboard … random variables –abstract concept probability...

Last Time

• Administrative Matters – Blackboard …

• Random Variables– Abstract concept

• Probability distribution Function– Summarizes probability structure– Sum to get any prob.

• Binomial Distribution

Reading In Textbook

Approximate Reading for Today’s Material:

Pages 311-317, 327-331, 372-375

Approximate Reading for Next Class:

Pages 377-381, 385-391, 488-491

Binomial Distribution

Setting: n independent trials of an

experiment with outcomes “Success” and

“Failure”, with P{S} = p.





Say X = #S’s has a “Binomial(n,p)

distribution”, and write “X ~ Bi(n,p)”





Say X = #S’s has a “Binomial(n,p)

distribution”, and write “X ~ Bi(n,p)”

• Called “parameters”

(really a family of distrib’ns, indexed by n & p)


E.g. Sampling with replacement

• “Experiment” is “draw a sample member”

• “S” is “vote for Candidate A”

• “p” is proportion in population for A

(note unknown, and goal of poll)

• Independent? (since with replacement)


E.g. Sampling with replacement

• “Experiment” is “draw a sample member”

• “S” is “vote for Candidate A”

• “p” is proportion in population for A

(note unknown, and goal of poll)

• Independent? (since with replacement)

X = #(for A) has a Binomial(n,p) dist’n


E.g. Sampling without replacement

• Draws are dependent

Result of 1st draw changes probs of 2nd draw

• P(S) on 2nd draw is no longer p

(again depends on 1st draw)

X = #(for A) is NOT Binomial


E.g. Sampling without replacement

• Draws are dependent

Result of 1st draw changes probs of 2nd draw

• P(S) on 2nd draw is no longer p

(again depends on 1st draw)

X = #(for A) is NOT Binomial

(although approximately true for large pop’n)


Models much more than political polls:

E.g. Coin tossing

(recall saw “independence” was good)

E.g. Shooting free throws (in basketball)

• Is p always the same?

• Really independent? (turns out to be OK)

Binomial Prob. Dist’n Func.

• Summarize all prob’s for X ~ Bi(n,p)



• By function: xXPxf



• By function:

Recall:

• Sum over this for any prob. about X

xXPxf



• By function:

Recall:

• Sum over this for any prob. about X

• Avoids doing complicated calculation each

time want a prob.

xXPxf


Repeat “experiment” (S or F) n times



• Outcomes “Success” or “Failure”



• Outcomes “Success” or “Failure”

• Independent repetitions

• Let X = # of S’s (count S’s)


Repeat (S or F) n times (ind.), let X = # of S’s



P[X = x] =

Desired probability distribution

function



P[X = x] =

Depends on particular draws,

So expand in those terms,

and use Big Rules of Probability



P[X = x] = P[(S1&…&Sx&Fx+1&…&Fn) or …]

• For “S on 1st draw”, “S on x-th draw”, …



P[X = x] = P[(S1&…&Sx&Fx+1&…&Fn) or …]


• One possible ordering of S,…,S,F,…,F

where: x of these

n-x of these



P[X = x] = P[(S1&…&Sx&Fx+1&…&Fn) or …]


• One possible ordering of S,…,S,F,…,F

• This includes all other orderings

(very many, but we can think of them)



P[X = x] = P[(S1&…&Sx&Fx+1&…&Fn) or …]

Next decompose with

and – or – not Rules of Probability



P[X = x] = P[(S1&…&Sx&Fx+1&…&Fn) or …] = =

P[(S1&…&Sx&Fx+1&…&Fn)] + …

• Disjoint OR rule [“or” add]



P[X = x] = P[(S1&…&Sx&Fx+1&…&Fn) or …] = =

P[(S1&…&Sx&Fx+1&…&Fn)] + …

• Disjoint OR rule [“or” add]

(recall “no overlap”)



P[X = x] = P[(S1&…&Sx&Fx+1&…&Fn) or …] = =

P[(S1&…&Sx&Fx+1&…&Fn)] + …

= P(S1)…P(Sx)P(Fx+1)…P(Fn) + …

• Independent AND rule [“and” mult.]



P[X = x] = P[(S1&…&Sx&Fx+1&…&Fn) or …] = =

P[(S1&…&Sx&Fx+1&…&Fn)] + …

= P(S1)…P(Sx)P(Fx+1)…P(Fn) + …

=

since p = P[S] since (1-p) = P[F]

xnx pp 1



P[X = x] = P[(S1&…&Sx&Fx+1&…&Fn) or …] = =

P[(S1&…&Sx&Fx+1&…&Fn)] + …

= P(S1)…P(Sx)P(Fx+1)…P(Fn) + …

=

since x = #S’s since (n-x) = #F’s

xnx pp 1



P[X = x] = P[(S1&…&Sx&Fx+1&…&Fn) or …] = =

P[(S1&…&Sx&Fx+1&…&Fn)] + …

= P(S1)…P(Sx)P(Fx+1)…P(Fn) + …

= = #(terms)

since all of these are the same, just count

xnx pp 1 xnx pp 1



P[X = x] = #(terms)

# ways to order S …S F …F

xnx pp 1



P[X = x] = #(terms)


Approach: have “n slots”

xnx pp 1



P[X = x] = #(terms)



“choose x of them to in which to put S”

xnx pp 1



P[X = x] = #(terms)



“choose x of them to in which to put S”

thus have #(terms) =

xnx pp 1

x

n



P[X = x] = #(terms)

=

general formula that works for all n, p, x

xnx pp 1

xnx ppx

n

1



P[X = x] = #(terms)

=

=

Binomial Probability Distribution Function

(for any n and p)

xnx pp 1

xnx ppx

n

1

xf



More complete representation

otherwise

nxppx

nxf

xnx

0

,...,01



More complete representation

But generally assume is understood, & write

otherwise

nxppx

nxf

xnx

0

,...,01

xnx ppx

nxf

1


Application of:

For X ~ Bi(n,p)

• Compute any probability for X

• By summing over appropriate values

xnx ppx

nxf

1

Application of Bi. Pro. Dist. Fun.

Application of:

E.g.: A system fails if any 3 of 5 independent

components fail

xnx ppx

nxf

1


Application of:


components fail

• Common setup in Reliability Theory

xnx ppx

nxf

1


Application of:


components fail

• Common setup in Reliability Theory

• Used when things “really need to work”

• E.g. aircraft components

xnx ppx

nxf

1


Application of:


components fail

If each component works 99% of time,

xnx ppx

nxf

1


Application of:


components fail

If each component works 99% of time,

how likely is the system to break down?

xnx ppx

nxf

1


Application of:

E.g.: Sys. F if 3 of 5 F, each works 99% time,


xnx ppx

nxf

1


Application of:



Let X = #F’s

xnx ppx

nxf

1


Application of:



Let X = #F’s, model X ~ Bi(5,0.01)

xnx ppx

nxf

1


Application of:




• Recall n = # of trials (repeats of experim’t)

xnx ppx

nxf

1


Application of:




• Components assumed independent

xnx ppx

nxf

1


Application of:




• Recall p = P(“S”), on each trial(works 99%, so fails 1%)

xnx ppx

nxf

1


Application of:




• Note S can in fact be “Failure of comp’t”

(opposite of usual intuition)

xnx ppx

nxf

1


Application of:




• Note S can in fact be “Failure of comp’t”

(it is just one outcome of exp’t)

xnx ppx

nxf

1


Application of:



P[system breaks down] = P[X ≥ 3]

recall X~Bi(5,0.01) counts failures

xnx ppx

nxf

1


Application of:



P[system breaks down] = P[X ≥ 3] =

(sum of prob. dist. func. over x ≥ 3)

xnx ppx

nxf

1

543 fff


Application of:




xnx ppx

nxf

1

543 fff

051423 99.001.05

599.001.0

4

599.001.0

3

5


Application of:




xnx ppx

nxf

1

543 fff

051423 99.001.05

599.001.0

4

599.001.0

3

5

610985.0


Application of:




Shows: great reliability

xnx ppx

nxf

1

543 fff

051423 99.001.05

599.001.0

4

599.001.0

3

5

610985.0


HW: C 12: A factory makes 10% defective

items & items are independently defective.

(maybe not great assumption, because many

causes of defects will give string of defects)




(maybe not great assumption, because many

causes of defects will give string of defects)

(but can call this an “approximate model”)




Find P{9 or more good items in 10}

a. Using X = # good items, and Binomial

probability distribution function. (0.736)

(Hint: consider “not” rule)







b. Using X = # bad items, and Binomial








b. Using X = # bad items, and Binomial


Note: will soon see easier way to do this, but

please use Bi. P. D. F. here

Research Corner

Medical Imaging – A Challenging ExampleMedical Imaging – A Challenging Example

Research Corner


• Male Pelvis• Bladder – Prostate – Rectum

Research Corner


• Male Pelvis• Bladder – Prostate – Rectum• How do they move over time (days)?• Critical to Radiation Treatment

(e.g. Prostate Cancer)

Research Corner


• Male Pelvis• Bladder – Prostate – Rectum• How do they move over time (days)?• Critical to Radiation Treatment

• Work with 3-d CT (“Computed Tomography”)

(3d version of Xray)

Research Corner


• Male Pelvis• Bladder – Prostate – Rectum• How do they move over time (days)?• Critical to Radiation Treatment Wo

• Work with 3-d CT• Very Challenging to “Segment”

• Find boundary of each object?• Represent each Object?

Male Pelvis – Raw DataMale Pelvis – Raw Data

One CT Slice

(in 3d image)

Coccyx

(Tail Bone)

Rectum

Bladder


Bladder:

manual segmentation

Slice by slice

Reassembled


Bladder:

Slices:

Reassembled in 3d

How to represent?

Thanks: Ja-Yeon Jeong

3-d m-reps3-d m-reps

Bladder – Prostate – Rectum (multiple objects, J. Y. Jeong)

• Medial Atoms provide “skeleton”

• Implied Boundary from “spokes” “surface”

Research Corner

How to understandHow to understand

““population level variation”?population level variation”?

Research Corner



Approach: Principal Geodesic AnalysisApproach: Principal Geodesic Analysis

• Focus on “modes of variation”Focus on “modes of variation”

Research Corner





• Ordered by “magnitude of variation”Ordered by “magnitude of variation”

Research Corner






• Need to “independent of each other”Need to “independent of each other”

Research Corner






• Need to “independent of each other”Need to “independent of each other”

(question for us: how to quantify?)(question for us: how to quantify?)

PGA for m-reps, Bladder-Prostate-Rectum

Bladder – Prostate – Rectum, 1 person, 17 days

PG 1 PG 2 PG 3

(analysis by Ja Yeon Jeong)


• Useful in many applications



• Have powerful method of calculation

Use Binomial probability dist’n function

& sum over needed values




• But a little painful to calculate

formula is involved (not easy hand calculation)

maybe very many terms (e.g. political polls)





• How about summaries?






Old Approach: Tables



Idea: somebody else calculates

“many Binomial probabilities”,

and stores results you can

look up:



In our Text: Table C




Note: Indexed by n

p

(recall Binomial is indexed family

of dist’ns)




Note: Indexed by n

p

and can input k (x) values

then read off P[X≤k]

Historical Note

• Tables were constructed well before

modern computers (1910s – 1930s)

Historical Note



• How was it done?

Historical Note



• How was it done?

Main Tool:

mechanical calculator

(hand powered)

(did repeated addition)

Historical Note

What was a “computer” in the early 1900s?

Historical Note


(the term did exist!)

Historical Note



A (human) job title!

Historical Note




Tables made by (carefully organized)

rooms full of people, all using mechanical

hand calculators

Historical Note




Tables made by (carefully organized)

rooms full of people, all using mechanical

hand calculators

Deep math was used for allocating resources






Modern Approach: Computers (electronic)






Modern Approach: Computers

In Excel: BINOMDIST function


Excel function: BINOMDIST



Access methods:



Access methods:

Generally in Excel:

Many ways to access things



Access methods:

1. Tool bar

– Click fx button



Access methods:

1. Tool bar

– Click fx button

– Pulls up function menu



Access methods:

1. Tool bar

– Click fx button


– Choose “statistical”



Access methods:

1. Tool bar

– Click fx button



– And BINOMDIST



Access methods:

1. Tool bar

– Click fx button



– And BINOMDIST

– Gives BINOMDIST menu



Access methods:

1. Tool bar

2. Formula Tab



Access methods:

1. Tool bar

2. Formula Tab

– More Functions



Access methods:

1. Tool bar

2. Formula Tab

– More Functions

– Statistical



Access methods:

1. Tool bar

2. Formula Tab

– More Functions

– Statistical

– BINOMDIST



Access methods:

1. Tool bar

2. Formula Tab

– More Functions

– Statistical

– BINOMDIST

Gets to same menu (as above)



Try these out, Class Example 2:http://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg2.xls

http://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg2.xls

Binomial Probs in EXCEL

To compute P{X=x}, for X ~ Bi(n,p):



Caution: Completely

different notation



x



x

n



x

n

p



Cumulative:

P{X=x}: false



Cumulative:

P{X=x}: false

P{X<=x}: true

(will illustrate soon)



Now check out specific problems,

Class Example 2:http://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg2.xls

http://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg2.xls


Class Example 2.1:

For X ~ Bi(1,0.5), i.e. toss a fair coin once,

count the number of Heads:


Class Example 2.1:



(a) "prob. of a Head" =

= P{X = 1}


Class Example 2.1:




= P{X = 1} =


Class Example 2.1:




= P{X = 1} =

= 0.5


Class Example 2.1:



(a) "prob. of a Head" = P{X = 1} = 0.5

Note: could also just

type formula in:


Class Example 2.1:



(a) "prob. of a Tail" =

= P{X = 0} =

= 0.5


Class Example 2.2:

For X ~ Bi(2,0.5), i.e. toss a fair coin twice,



Class Example 2.2:



(a) "prob. of no Heads" =

= P{X = 0} =


Class Example 2.2:




= P{X = 0} =

= P{T1 and T2}

= P{T1}*P{T2} = 0.25


Class Example 2.2:



(b) "prob. of one Head" =

= P{X = 1} =

(harder calculation)


Class Example 2.3:

For X ~ Bi(2,0.3), i.e. toss an unbalanced coin

twice, count the number of Heads:


Class Example 2.3:




= P{X = 0} =

= P{T1 and T2} =

= P{T1}*P{T2} = 0.49


Class Example 2.3:



(b) "prob. of one Head" =

= P{X = 1} =


Class Example 2.4:

For X ~ Bi(20,0.3), i.e. toss an unbalanced

coin 20 times, count the number of Heads:


Class Example 2.4:




= P{X = 0} =


Class Example 2.4:




= P{X = 0} =

= 0.000797923


Class Example 2.4:




= P{X = 0} =

= 0.000797923

Check: 0.7^20 = 0.000797923


Class Example 2.4:



(c) "prob. of six Heads" =

= P{X = 6} =


Class Example 2.4:



(d) "prob. of at most 6 Heads" =

= P{X ≤ 6}


Class Example 2.4:




= P{X ≤ 6}

Solution 1: Add them up


Class Example 2.4:




= P{X ≤ 6}

Solution 1: Add them up

= 0.60801


Class Example 2.4:




= P{X ≤ 6}

Solution 1: Add, = 0.60801


Class Example 2.4:




= P{X ≤ 6}


Solution 2: Use Cumulative


Class Example 2.4:




= P{X ≤ 6}


Solution 2: Use Cumulative

Same answer


Class Example 2.4: For X ~ Bi(20,0.3) :

(e) "prob. of at least 6 Heads" = P{X ≥ 6}



(e) "prob. of at least 6 Heads" = P{X ≥ 6}

Caution: cumulative works "other way", so

need to put in Excel usable form



(e) "prob. of at least 6 Heads" = P{X ≥ 6} =

4 5 6 7




= 1 - P{not X ≥ 6} =

4 5 6 7




= 1 - P{not X ≥ 6} =

= 1 - P{X < 6}

4 5 6 7




= 1 - P{not X ≥ 6} =

= 1 - P{X < 6} = 1 - P{X ≤ 5} (since

counting

4 5 6 7 numbers)




= 1 - P{not X ≥ 6} =

= 1 - P{X < 6} = 1 - P{X ≤ 5}

Now use BINOMDIST & Cumulative = true

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