Upload
shanna-conley
View
214
Download
1
Embed Size (px)
Citation preview
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