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.

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)”

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)”

• Called “parameters”

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

Binomial Distribution

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)

Binomial Distribution

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

Binomial Distribution

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

Binomial Distribution

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)

Binomial Distribution

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)

Binomial Prob. Dist’n Func.

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

• By function: xXPxf

Binomial Prob. Dist’n Func.

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

• By function:

Recall:

• Sum over this for any prob. about X

xXPxf

Binomial Prob. Dist’n Func.

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

• By function:

Recall:

• Sum over this for any prob. about X

• Avoids doing complicated calculation each

time want a prob.

xXPxf

Binomial Prob. Dist’n Func.

Repeat “experiment” (S or F) n times

Binomial Prob. Dist’n Func.

Repeat “experiment” (S or F) n times

• Outcomes “Success” or “Failure”

Binomial Prob. Dist’n Func.

Repeat “experiment” (S or F) n times

• Outcomes “Success” or “Failure”

• Independent repetitions

• Let X = # of S’s (count S’s)

Binomial Prob. Dist’n Func.

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

Binomial Prob. Dist’n Func.

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

P[X = x] =

Desired probability distribution

function

Binomial Prob. Dist’n Func.

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

P[X = x] =

Depends on particular draws,

So expand in those terms,

and use Big Rules of Probability

Binomial Prob. Dist’n Func.

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

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

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

Binomial Prob. Dist’n Func.

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

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

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

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

where: x of these

n-x of these

Binomial Prob. Dist’n Func.

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

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

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

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

• This includes all other orderings

(very many, but we can think of them)

Binomial Prob. Dist’n Func.

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

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

Next decompose with

and – or – not Rules of Probability

Binomial Prob. Dist’n Func.

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

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

P[(S1&…&Sx&Fx+1&…&Fn)] + …

• Disjoint OR rule [“or” add]

Binomial Prob. Dist’n Func.

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

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

P[(S1&…&Sx&Fx+1&…&Fn)] + …

• Disjoint OR rule [“or” add]

(recall “no overlap”)

Binomial Prob. Dist’n Func.

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

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.]

Binomial Prob. Dist’n Func.

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

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

Binomial Prob. Dist’n Func.

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

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

Binomial Prob. Dist’n Func.

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

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

Binomial Prob. Dist’n Func.

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

P[X = x] = #(terms)

# ways to order S …S F …F

xnx pp 1

Binomial Prob. Dist’n Func.

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

P[X = x] = #(terms)

# ways to order S …S F …F

Approach: have “n slots”

xnx pp 1

Binomial Prob. Dist’n Func.

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

P[X = x] = #(terms)

# ways to order S …S F …F

Approach: have “n slots”

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

xnx pp 1

Binomial Prob. Dist’n Func.

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

P[X = x] = #(terms)

# ways to order S …S F …F

Approach: have “n slots”

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

thus have #(terms) =

xnx pp 1

x

n

Binomial Prob. Dist’n Func.

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

P[X = x] = #(terms)

=

general formula that works for all n, p, x

xnx pp 1

xnx ppx

n

1

Binomial Prob. Dist’n Func.

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

P[X = x] = #(terms)

=

=

Binomial Probability Distribution Function

(for any n and p)

xnx pp 1

xnx ppx

n

1

xf

Binomial Prob. Dist’n Func.

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

More complete representation

otherwise

nxppx

nxf

xnx

0

,...,01

Binomial Prob. Dist’n Func.

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

More complete representation

But generally assume is understood, & write

otherwise

nxppx

nxf

xnx

0

,...,01

xnx ppx

nxf

1

Binomial Prob. Dist’n Func.

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 Bi. Pro. Dist. Fun.

Application of:

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

components fail

• Common setup in Reliability Theory

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

• Common setup in Reliability Theory

• Used when things “really need to work”

• E.g. aircraft components

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

If each component works 99% of time,

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

If each component works 99% of time,

how likely is the system to break down?

xnx ppx

nxf

1

Application of Bi. Pro. Dist. Fun.

Application of:

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

how likely is the system to break down?

xnx ppx

nxf

1

Application of Bi. Pro. Dist. Fun.

Application of:

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

how likely is the system to break down?

Let X = #F’s

xnx ppx

nxf

1

Application of Bi. Pro. Dist. Fun.

Application of:

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

how likely is the system to break down?

Let X = #F’s, model X ~ Bi(5,0.01)

xnx ppx

nxf

1

Application of Bi. Pro. Dist. Fun.

Application of:

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

how likely is the system to break down?

Let X = #F’s, model X ~ Bi(5,0.01)

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

xnx ppx

nxf

1

Application of Bi. Pro. Dist. Fun.

Application of:

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

how likely is the system to break down?

Let X = #F’s, model X ~ Bi(5,0.01)

• Components assumed independent

xnx ppx

nxf

1

Application of Bi. Pro. Dist. Fun.

Application of:

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

how likely is the system to break down?

Let X = #F’s, model X ~ Bi(5,0.01)

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

xnx ppx

nxf

1

Application of Bi. Pro. Dist. Fun.

Application of:

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

how likely is the system to break down?

Let X = #F’s, model X ~ Bi(5,0.01)

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

(opposite of usual intuition)

xnx ppx

nxf

1

Application of Bi. Pro. Dist. Fun.

Application of:

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

how likely is the system to break down?

Let X = #F’s, model X ~ Bi(5,0.01)

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

(it is just one outcome of exp’t)

xnx ppx

nxf

1

Application of Bi. Pro. Dist. Fun.

Application of:

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

how likely is the system to break down?

P[system breaks down] = P[X ≥ 3]

recall X~Bi(5,0.01) counts failures

xnx ppx

nxf

1

Application of Bi. Pro. Dist. Fun.

Application of:

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

how likely is the system to break down?

P[system breaks down] = P[X ≥ 3] =

(sum of prob. dist. func. over x ≥ 3)

xnx ppx

nxf

1

543 fff

Application of Bi. Pro. Dist. Fun.

Application of:

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

how likely is the system to break down?

P[system breaks down] = P[X ≥ 3] =

xnx ppx

nxf

1

543 fff

051423 99.001.05

599.001.0

4

599.001.0

3

5

Application of Bi. Pro. Dist. Fun.

Application of:

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

how likely is the system to break down?

P[system breaks down] = P[X ≥ 3] =

xnx ppx

nxf

1

543 fff

051423 99.001.05

599.001.0

4

599.001.0

3

5

610985.0

Application of Bi. Pro. Dist. Fun.

Application of:

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

how likely is the system to break down?

P[system breaks down] = P[X ≥ 3] =

Shows: great reliability

xnx ppx

nxf

1

543 fff

051423 99.001.05

599.001.0

4

599.001.0

3

5

610985.0

Application of Bi. Pro. Dist. Fun.

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)

Application of Bi. Pro. Dist. Fun.

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)

(but can call this an “approximate model”)

Application of Bi. Pro. Dist. Fun.

HW: C 12: A factory makes 10% defective

items & items are independently defective.

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)

Application of Bi. Pro. Dist. Fun.

HW: C 12: A factory makes 10% defective

items & items are independently defective.

Find P{9 or more good items in 10}

a. Using X = # good items, and Binomial

probability distribution function. (0.736)

b. Using X = # bad items, and Binomial

probability distribution function. (0.736)

Application of Bi. Pro. Dist. Fun.

HW: C 12: A factory makes 10% defective

items & items are independently defective.

Find P{9 or more good items in 10}

a. Using X = # good items, and Binomial

probability distribution function. (0.736)

b. Using X = # bad items, and Binomial

probability distribution function. (0.736)

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

Medical Imaging – A Challenging ExampleMedical Imaging – A Challenging Example

• Male Pelvis• Bladder – Prostate – Rectum

Research Corner

Medical Imaging – A Challenging ExampleMedical Imaging – A Challenging Example

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

(e.g. Prostate Cancer)

Research Corner

Medical Imaging – A Challenging ExampleMedical Imaging – A Challenging Example

• 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

Medical Imaging – A Challenging ExampleMedical Imaging – A Challenging Example

• 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?

Research Corner

Medical Imaging – A Challenging ExampleMedical Imaging – A Challenging Example

• 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

Male Pelvis – Raw DataMale Pelvis – Raw Data

Bladder:

manual segmentation

Slice by slice

Reassembled

Male Pelvis – Raw DataMale Pelvis – Raw Data

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

How to understandHow to understand

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

Approach: Principal Geodesic AnalysisApproach: Principal Geodesic Analysis

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

Research Corner

How to understandHow to understand

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

Approach: Principal Geodesic AnalysisApproach: Principal Geodesic Analysis

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

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

Research Corner

How to understandHow to understand

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

Approach: Principal Geodesic AnalysisApproach: Principal Geodesic Analysis

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

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

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

Research Corner

How to understandHow to understand

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

Approach: Principal Geodesic AnalysisApproach: Principal Geodesic Analysis

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

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

• 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)

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)

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)

Binomial Distribution

• Useful in many applications

Binomial Distribution

• Useful in many applications

• Have powerful method of calculation

Use Binomial probability dist’n function

& sum over needed values

Binomial Distribution

• Useful in many applications

• Have powerful method of calculation

• But a little painful to calculate

formula is involved (not easy hand calculation)

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

Binomial Distribution

• Useful in many applications

• Have powerful method of calculation

• But a little painful to calculate

• How about summaries?

Binomial Distribution

• Useful in many applications

• Have powerful method of calculation

• But a little painful to calculate

• How about summaries?

Old Approach: Tables

Binomial Distribution

Old Approach: Tables

Idea: somebody else calculates

“many Binomial probabilities”,

and stores results you can

look up:

Binomial Distribution

Old Approach: Tables

In our Text: Table C

Binomial Distribution

Old Approach: Tables

In our Text: Table C

Note: Indexed by n

p

(recall Binomial is indexed family

of dist’ns)

Binomial Distribution

Old Approach: Tables

In our Text: Table C

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

• Tables were constructed well before

modern computers (1910s – 1930s)

• How was it done?

Historical Note

• Tables were constructed well before

modern computers (1910s – 1930s)

• 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

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

(the term did exist!)

Historical Note

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

(the term did exist!)

A (human) job title!

Historical Note

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

(the term did exist!)

A (human) job title!

Tables made by (carefully organized)

rooms full of people, all using mechanical

hand calculators

Historical Note

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

(the term did exist!)

A (human) job title!

Tables made by (carefully organized)

rooms full of people, all using mechanical

hand calculators

Deep math was used for allocating resources

Binomial Distribution

• Useful in many applications

• Have powerful method of calculation

• But a little painful to calculate

• How about summaries?

Modern Approach: Computers (electronic)

Binomial Distribution

• Useful in many applications

• Have powerful method of calculation

• But a little painful to calculate

• How about summaries?

Modern Approach: Computers

In Excel: BINOMDIST function

Binomial Distribution

Excel function: BINOMDIST

Binomial Distribution

Excel function: BINOMDIST

Access methods:

Binomial Distribution

Excel function: BINOMDIST

Access methods:

Generally in Excel:

Many ways to access things

Binomial Distribution

Excel function: BINOMDIST

Access methods:

1. Tool bar

– Click fx button

Binomial Distribution

Excel function: BINOMDIST

Access methods:

1. Tool bar

– Click fx button

– Pulls up function menu

Binomial Distribution

Excel function: BINOMDIST

Access methods:

1. Tool bar

– Click fx button

– Pulls up function menu

– Choose “statistical”

Binomial Distribution

Excel function: BINOMDIST

Access methods:

1. Tool bar

– Click fx button

– Pulls up function menu

– Choose “statistical”

– And BINOMDIST

Binomial Distribution

Excel function: BINOMDIST

Access methods:

1. Tool bar

– Click fx button

– Pulls up function menu

– Choose “statistical”

– And BINOMDIST

– Gives BINOMDIST menu

Binomial Distribution

Excel function: BINOMDIST

Access methods:

1. Tool bar

2. Formula Tab

Binomial Distribution

Excel function: BINOMDIST

Access methods:

1. Tool bar

2. Formula Tab

– More Functions

Binomial Distribution

Excel function: BINOMDIST

Access methods:

1. Tool bar

2. Formula Tab

– More Functions

– Statistical

Binomial Distribution

Excel function: BINOMDIST

Access methods:

1. Tool bar

2. Formula Tab

– More Functions

– Statistical

– BINOMDIST

Binomial Distribution

Excel function: BINOMDIST

Access methods:

1. Tool bar

2. Formula Tab

– More Functions

– Statistical

– BINOMDIST

Gets to same menu (as above)

Binomial Distribution

Excel function: BINOMDIST

Try these out, Class Example 2: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):

Binomial Probs in EXCEL

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

Caution: Completely

different notation

Binomial Probs in EXCEL

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

x

Binomial Probs in EXCEL

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

x

n

Binomial Probs in EXCEL

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

x

n

p

Binomial Probs in EXCEL

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

Cumulative:

P{X=x}: false

Binomial Probs in EXCEL

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

Cumulative:

P{X=x}: false

P{X<=x}: true

(will illustrate soon)

Binomial Distribution

Excel function: BINOMDIST

Now check out specific problems,

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

Binomial Distribution

Class Example 2.1:

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

count the number of Heads:

Binomial Distribution

Class Example 2.1:

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

count the number of Heads:

(a) "prob. of a Head" =

= P{X = 1}

Binomial Distribution

Class Example 2.1:

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

count the number of Heads:

(a) "prob. of a Head" =

= P{X = 1} =

Binomial Distribution

Class Example 2.1:

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

count the number of Heads:

(a) "prob. of a Head" =

= P{X = 1} =

= 0.5

Binomial Distribution

Class Example 2.1:

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

count the number of Heads:

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

Note: could also just

type formula in:

Binomial Distribution

Class Example 2.1:

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

count the number of Heads:

(a) "prob. of a Tail" =

= P{X = 0} =

= 0.5

Binomial Distribution

Class Example 2.2:

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

count the number of Heads:

Binomial Distribution

Class Example 2.2:

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

count the number of Heads:

(a) "prob. of no Heads" =

= P{X = 0} =

Binomial Distribution

Class Example 2.2:

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

count the number of Heads:

(a) "prob. of no Heads" =

= P{X = 0} =

Binomial Distribution

Class Example 2.2:

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

count the number of Heads:

(a) "prob. of no Heads" =

= P{X = 0} =

= P{T1 and T2}

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

Binomial Distribution

Class Example 2.2:

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

count the number of Heads:

(b) "prob. of one Head" =

= P{X = 1} =

(harder calculation)

Binomial Distribution

Class Example 2.3:

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

twice, count the number of Heads:

Binomial Distribution

Class Example 2.3:

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

twice, count the number of Heads:

(a) "prob. of no Heads" =

= P{X = 0} =

= P{T1 and T2} =

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

Binomial Distribution

Class Example 2.3:

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

twice, count the number of Heads:

(b) "prob. of one Head" =

= P{X = 1} =

Binomial Distribution

Class Example 2.4:

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

coin 20 times, count the number of Heads:

Binomial Distribution

Class Example 2.4:

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

coin 20 times, count the number of Heads:

(a) "prob. of no Heads" =

= P{X = 0} =

Binomial Distribution

Class Example 2.4:

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

coin 20 times, count the number of Heads:

(a) "prob. of no Heads" =

= P{X = 0} =

= 0.000797923

Binomial Distribution

Class Example 2.4:

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

coin 20 times, count the number of Heads:

(a) "prob. of no Heads" =

= P{X = 0} =

= 0.000797923

Check: 0.7^20 = 0.000797923

Binomial Distribution

Class Example 2.4:

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

coin 20 times, count the number of Heads:

(c) "prob. of six Heads" =

= P{X = 6} =

Binomial Distribution

Class Example 2.4:

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

coin 20 times, count the number of Heads:

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

= P{X ≤ 6}

Binomial Distribution

Class Example 2.4:

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

coin 20 times, count the number of Heads:

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

= P{X ≤ 6}

Solution 1: Add them up

Binomial Distribution

Class Example 2.4:

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

coin 20 times, count the number of Heads:

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

= P{X ≤ 6}

Solution 1: Add them up

= 0.60801

Binomial Distribution

Class Example 2.4:

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

coin 20 times, count the number of Heads:

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

= P{X ≤ 6}

Solution 1: Add, = 0.60801

Binomial Distribution

Class Example 2.4:

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

coin 20 times, count the number of Heads:

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

= P{X ≤ 6}

Solution 1: Add, = 0.60801

Solution 2: Use Cumulative

Binomial Distribution

Class Example 2.4:

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

coin 20 times, count the number of Heads:

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

= P{X ≤ 6}

Solution 1: Add, = 0.60801

Solution 2: Use Cumulative

Binomial Distribution

Class Example 2.4:

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

coin 20 times, count the number of Heads:

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

= P{X ≤ 6}

Solution 1: Add, = 0.60801

Solution 2: Use Cumulative

Same answer

Binomial Distribution

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

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

Binomial Distribution

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

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

Caution: cumulative works "other way", so

need to put in Excel usable form

Binomial Distribution

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

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

4 5 6 7

Binomial Distribution

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

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

= 1 - P{not X ≥ 6} =

4 5 6 7

Binomial Distribution

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

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

= 1 - P{not X ≥ 6} =

= 1 - P{X < 6}

4 5 6 7

Binomial Distribution

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

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

= 1 - P{not X ≥ 6} =

= 1 - P{X < 6} = 1 - P{X ≤ 5} (since

counting

4 5 6 7 numbers)

Binomial Distribution

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

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

= 1 - P{not X ≥ 6} =

= 1 - P{X < 6} = 1 - P{X ≤ 5}

Now use BINOMDIST & Cumulative = true