Last Time Administrative Matters – Blackboard … Random Variables –Abstract concept Probability distribution Function –Summarizes probability structure.

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 1 st draw changes probs of 2 nd draw P(S) on 2 nd draw is no longer p (again depends on 1 st draw) X = #(for A) is NOT Binomial

Binomial Distribution E.g. Sampling without replacement Draws are dependent Result of 1 st draw changes probs of 2 nd draw P(S) on 2 nd draw is no longer p (again depends on 1 st 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:

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

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.

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[(S 1 &…&S x &F x+1 &…&F n ) or …] For “S on 1 st 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[(S 1 &…&S x &F x+1 &…&F n ) or …] For “S on 1 st 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[(S 1 &…&S x &F x+1 &…&F n ) or …] For “S on 1 st 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[(S 1 &…&S x &F x+1 &…&F n ) 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[(S 1 &…&S x &F x+1 &…&F n ) or …] = = P[(S 1 &…&S x &F x+1 &…&F n )] + … 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[(S 1 &…&S x &F x+1 &…&F n ) or …] = = P[(S 1 &…&S x &F x+1 &…&F n )] + … 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[(S 1 &…&S x &F x+1 &…&F n ) or …] = = P[(S 1 &…&S x &F x+1 &…&F n )] + … = P(S 1 )…P(S x )P(F x+1 )…P(F n ) + … 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[(S 1 &…&S x &F x+1 &…&F n ) or …] = = P[(S 1 &…&S x &F x+1 &…&F n )] + … = P(S 1 )…P(S x )P(F x+1 )…P(F n ) + … = since p = P[S] since (1-p) = P[F]

Binomial Prob. Dist’n Func. Repeat (S or F) n times (ind.), let X = # of S’s P[X = x] = P[(S 1 &…&S x &F x+1 &…&F n ) or …] = = P[(S 1 &…&S x &F x+1 &…&F n )] + … = P(S 1 )…P(S x )P(F x+1 )…P(F n ) + … = since x = #S’s since (n-x) = #F’s

Binomial Prob. Dist’n Func. Repeat (S or F) n times (ind.), let X = # of S’s P[X = x] = P[(S 1 &…&S x &F x+1 &…&F n ) or …] = = P[(S 1 &…&S x &F x+1 &…&F n )] + … = P(S 1 )…P(S x )P(F x+1 )…P(F n ) + … = = #(terms) since all of these are the same, just count

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

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”

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”

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) =

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

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)

Binomial Prob. Dist’n Func. Repeat (S or F) n times (ind.), let X = # of S’s More complete representation

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

Binomial Prob. Dist’n Func. Application of: For X ~ Bi(n,p) Compute any probability for X By summing over appropriate values

Application of Bi. Pro. Dist. Fun. Application of: E.g.: A system fails if any 3 of 5 independent components fail

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

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

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,

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?

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?

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

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)

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)

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

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%)

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)

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)

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

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)

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] =

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

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 Example

Research Corner Medical Imaging – A Challenging Example Male Pelvis Bladder – Prostate – Rectum

Research Corner Medical 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 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 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 Data One CT Slice (in 3d image) Coccyx (Tail Bone) Rectum Bladder

Male Pelvis – Raw Data Bladder: manual segmenta tion Slice by slice Reassembled

Male Pelvis – Raw Data Bladder: Slices: Reassembled in 3d How to represent? Thanks: Ja-Yeon Jeong

3-d m-reps Bladder – Prostate – Rectum (multiple objects, J. Y. Jeong) Medial Atoms provide “skeleton” Implied Boundary from “spokes”  “surface”

Research Corner How to understand “population level variation”?

Research Corner How to understand “population level variation”? Approach: Principal Geodesic Analysis Focus on “modes of variation”Focus on “modes of variation”

Research Corner How to understand “population level variation”? Approach: 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 understand “population level variation”? Approach: 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 understand “population level variation”? Approach: 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?)

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 f x button

Binomial Distribution Excel function: BINOMDIST Access methods: 1.Tool bar –Click f x button –Pulls up function menu

Binomial Distribution Excel function: BINOMDIST Access methods: 1.Tool bar –Click f x button –Pulls up function menu –Choose “statistical”

Binomial Distribution Excel function: BINOMDIST Access methods: 1.Tool bar –Click f x button –Pulls up function menu –Choose “statistical” –And BINOMDIST

Binomial Distribution Excel function: BINOMDIST Access methods: 1.Tool bar –Click f x 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} = = P{T 1 and T 2 } = P{T 1 }*P{T 2 } = 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{T 1 and T 2 } = = P{T 1 }*P{T 2 } = 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 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

Last Time Administrative Matters – Blackboard … Random Variables –Abstract concept Probability distribution Function –Summarizes probability structure.

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Last Time Administrative Matters – Blackboard … Random Variables –Abstract concept Probability distribution Function –Summarizes probability structure.

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