Math 1710 Class 8
Math 1710Class 8V2BinomialTwo Ways of“Randomly”Flipping 2CoinsSobrietyCheckpointsNormalDistributionWorking rmalApproximationAccurateMath 1710 Class 8Dr. Allen BackSep. 12, 2016
Three Girls Out of Five ChildrenMath 1710Class 8V2BinomialTwo Ways of“Randomly”Flipping 2CoinsSobrietyCheckpointsNormalDistributionWorking rmalApproximationAccurateSuppose P(Girl) .6 and gender of births independent.
Three Girls Out of Five ChildrenMath 1710Class 8V2BinomialTwo Ways of“Randomly”Flipping 2CoinsSobrietyCheckpointsNormalDistributionWorking rmalApproximationAccurateSuppose P(Girl) .6 and gender of births independent.P(3 Girls)?
Three Girls Out of Five ChildrenMath 1710Class 8V2BinomialTwo Ways of“Randomly”Flipping 2CoinsSobrietyCheckpointsNormalDistributionWorking rmalApproximationAccurateSuppose P(Girl) .6 and gender of births independent.P(3 Girls)?P(first 3 G, last 2 B) ?
Three Girls Out of Five ChildrenMath 1710Class 8V2BinomialTwo Ways of“Randomly”Flipping 2CoinsSobrietyCheckpointsNormalDistributionWorking rmalApproximationAccurateSuppose P(Girl) .6 and gender of births independent.P(3 Girls)?P(first 3 G, last 2 B) P(GGGBB) .63 .42 .03456.But this undercounts the answer.
Three Girls Out of Five ChildrenMath 1710Class 8V2BinomialTwo Ways of“Randomly”Flipping 2CoinsSobrietyCheckpointsNormalDistributionWorking rmalApproximationAccurateSuppose P(Girl) .6 and gender of births independent.P(3 Girls)?P(first 3 G, last 2 B) P(GGGBB) .63 .42 .03456.But this undercounts the answer.Other orders also possible; e.g. BBGGG .
Three Girls Out of Five ChildrenMath 1710Class 8V2BinomialTwo Ways of“Randomly”Flipping 2CoinsSobrietyCheckpointsNormalDistributionWorking rmalApproximationAccurateSuppose P(Girl) .6 and gender of births independent.P(3 Girls)?P(first 3 G, last 2 B) P(GGGBB) .63 .42 .03456.But this undercounts the answer.Other orders also possible; e.g. BBGGG .How many such orders?
Three Girls Out of Five ChildrenMath 1710Class 8V2BinomialTwo Ways of“Randomly”Flipping 2CoinsSobrietyCheckpointsNormalDistributionWorking rmalApproximationAccurateSuppose P(Girl) .6 and gender of births independent.P(3 Girls)?P(first 3 G, last 2 B) P(GGGBB) .63 .42 .03456.But this undercounts the answer.10 Possible Orders:First child a girl: GGGBB GGBGB GGBBG GBGGBGBGBG GBBGGFirst child a boy: BGGGB BGGBG BGBGG BBGGG
Three Girls Out of Five ChildrenMath 1710Class 8V2BinomialTwo Ways of“Randomly”Flipping 2CoinsSobrietyCheckpointsNormalDistributionWorking rmalApproximationAccurateSuppose P(Girl) .6 and gender of births independent.P(3 Girls)?P(first 3 G, last 2 B) P(GGGBB) .63 .42 .03456.But this undercounts the answer.10 Possible Orders:First child a girl: GGGBB GGBGB GGBBG GBGGBGBGBG GBBGGFirst child a boy: BGGGB BGGBG BGBGG BBGGGSo answer is 10(.6)3 (.4)2 .3456.
Why 10 Possible Orders?Math 1710Class 8V2BinomialTwo Ways of“Randomly”Flipping 2CoinsSobrietyCheckpointsNormalDistributionWorking rmalApproximationAccurate5 birth positions, 3 of which girls
Why 10 Possible Orders?Math 1710Class 8V2BinomialTwo Ways of“Randomly”Flipping 2CoinsSobrietyCheckpointsNormalDistributionWorking rmalApproximationAccurate5 birth positions, 3 of which girlsSo 5C5,3 3
Why 10 Possible Orders?Math 1710Class 8V2BinomialTwo Ways of“Randomly”Flipping 2CoinsSobrietyCheckpointsNormalDistributionWorking rmalApproximationAccurate5 birth positions, 3 of which girls 5·4·35C5,3 31·2·3
Why 10 Possible Orders?Math 1710Class 8V2BinomialTwo Ways of“Randomly”Flipping 2CoinsSobrietyCheckpointsNormalDistributionWorking rmalApproximationAccurate5 birth positions, 3 of which girls 5·4·35·45 1031·2·31·2
Why 10 Possible Orders?Math 1710Class 8V2BinomialTwo Ways of“Randomly”Flipping 2CoinsSobrietyCheckpointsNormalDistributionWorking rmalApproximationAccurate5 birth positions, 3 of which girls 5·4·35·45 1031·2·31·2Above showed 55 32which works in general as well.
Why 10 Possible Orders?Math 1710Class 8V2BinomialTwo Ways of“Randomly”Flipping 2CoinsSobrietyCheckpoints5 birth positions, 3 of which girls 5·45·4·35 10 31·2·31·2If one tacks a 2! onto both the numerator and denominator,5·4·35·4·3·2·1 1·2·33!2!NormalDistributionWorking rmalApproximationAccurateshowing 5!5 33!2!which is also a general formula.
Why C5,3 5·4·31·2·3 ?Math 1710Class 8V2BinomialTwo Ways of“Randomly”Flipping 2CoinsSobrietyCheckpointsNormalDistributionWorking rmalApproximationAccurate5 birth positions, 3 of which girls
Why C5,3 5·4·31·2·3 ?Math 1710Class 8V2BinomialTwo Ways of“Randomly”Flipping 2CoinsSobrietyCheckpointsNormalDistributionWorking rmalApproximationAccurate5 birth positions, 3 of which girlsSuppose the three girls are named Abby, Betty, and Carla.
Why C5,3 5·4·31·2·3 ?Math 1710Class 8V2BinomialTwo Ways of“Randomly”Flipping 2CoinsSobrietyCheckpointsNormalDistributionWorking rmalApproximationAccurate5 birth positions, 3 of which girlsSuppose the three girls are named Abby, Betty, and Carla.Then 5 birth positions for Abby.4 for Betty.3 for Carla.
Why C5,3 5·4·31·2·3 ?Math 1710Class 8V2BinomialTwo Ways of“Randomly”Flipping 2CoinsSobrietyCheckpointsNormalDistributionWorking rmalApproximationAccurate5 birth positions, 3 of which girlsSuppose the three girls are named Abby, Betty, and Carla.Then 5 birth positions for Abby.4 for Betty.3 for Carla.But each set of 3 birth positions for the girls shows up 3! timesdepending on the order of births among Abby, Betty, and Carla.
Why C5,3 5·4·31·2·3 ?Math 1710Class 8V2NormalDistribution5 birth positions, 3 of which girlsSuppose the three girls are named Abby, Betty, and Carla.Then 5 birth positions for Abby.4 for Betty.3 for Carla.But each set of 3 birth positions for the girls shows up 3! timesdepending on the order of births among Abby, Betty, and Carla.Working WithNormalDistributionsSoBinomialTwo Ways of“Randomly”Flipping NormalApproximationAccurateC5,3 5·4·35 31·2·3
Pascal’s TriangleMath 1710Class 8V21Binomial1Two Ways of“Randomly”Flipping king rmalApproximationAccurate(a b)0(a b)1(a b)2(a b)3.23451136101410 1 1a 1b 1a2 2ab 1b 2 . 1a3 3a2 b 3ab 2 1b 3 .151
Pascal’s TriangleMath 1710Class 8V21Binomial1Two Ways of“Randomly”Flipping ng rmalApproximationAccurate113451213610141015110 Possible Orders: (Two points of view.)First child a girl: GGGBB GGBGB GGBBG GBGGBGBGBG GBBGGFirst child a boy: BGGGB BGGBG BGBGG BBGGG
Pascal’s TriangleMath 1710Class 81V21Binomial1Two Ways of“Randomly”Flipping 2CoinsSobrietyCheckpointsNormalDistributionWorking rmalApproximationAccurate1113451213101461015110 Possible Orders: (Three points of view.)First child a girl: GGGBB GGBGB GGBBG GBGGBGBGBG GBBGGFirst child a boy: BGGGB BGGBG BGBGG BBGGG 544 .323
General TerminologyMath 1710Class 8V2BinomialTwo Ways of“Randomly”Flipping 2CoinsSobrietyCheckpointsNormalDistributionWorking rmalApproximationAccurateIn general we speak about a sequence of Bernoulli Trials:2 outcomes, conventionally called success and failure.constant probablility p of success.the successive trials are independent.So, for each trial, the number of successes (0 or 1) is aBernoulli(p) RV.
General TerminologyMath 1710Class 8V2BinomialTwo Ways of“Randomly”Flipping 2CoinsBinomial(n,p) RV Y describes the number of successes in nBernoulli trials. For Y we know µ np, σ npq, andSobrietyCheckpoints n k n kP(Y k) p q.kNormalDistributionWorking rmalApproximationAccurate(Here q 1 p.)
General TerminologyMath 1710Class 8V2BinomialTwo Ways of“Randomly”Flipping 2CoinsSobrietyCheckpointsNormalDistributionWorking rmalApproximationAccurateBinomial(n,p) RV Y describes the number of successes in nBernoulli trials. For Y we know µ np, σ npq, and n k n kP(Y k) p q.k(Here q 1 p.)A substitute for a big table giving the prob. dist. of Y .
General TerminologyMath 1710Class 8V2BinomialTwo Ways of“Randomly”Flipping 2CoinsSobrietyCheckpointsNormalDistributionWorking rmalApproximationAccurateWhy µ np, σ npq for a Binomial(n,p) RV Y ?
General TerminologyMath 1710Class 8V2BinomialTwo Ways of“Randomly”Flipping 2CoinsSobrietyCheckpointsNormalDistributionWorking rmalApproximationAccurate Why µ np, σ npq for a Binomial(n,p) RV Y ?First confirm for a Bernoulli(p) RV, µ p and the the varianceis pq.
General TerminologyMath 1710Class 8V2BinomialTwo Ways of“Randomly”Flipping 2CoinsSobrietyCheckpointsNormalDistributionWorking rmalApproximationAccurate Why µ np, σ npq for a Binomial(n,p) RV Y ?Now let Xi be a Bernoulli(p) RV counting the number ofsuccesses (1 or 0) on the i’th trial.
General TerminologyMath 1710Class 8V2BinomialTwo Ways of“Randomly”Flipping 2CoinsSobrietyCheckpointsWhy µ np, σ npq for a Binomial(n,p) RV Y ?Y X1 X2 . . . Xn .NormalDistributionWorking rmalApproximationAccurate(Remember Y is the total number of successes.)
General TerminologyMath 1710Class 8V2BinomialTwo Ways of“Randomly”Flipping 2CoinsSobrietyCheckpointsNormalDistributionWorking rmalApproximationAccurateWhy µ np, σ npq for a Binomial(n,p) RV Y ?Y X1 X2 . . . Xn .(Remember Y is the total number of successes.)Since both means and variances add for the sum of independentRV’s, we obtain the formulas for the binomial case.
Suppose 70% approve the President . . .Math 1710Class 8V2BinomialTwo Ways of“Randomly”Flipping 2CoinsSobrietyCheckpointsNormalDistributionWorking rmalApproximationAccurateYou poll 100 people.What is the probability that exactly 65 report approval?
Suppose 70% approve the President . . .Math 1710Class 8V2BinomialTwo Ways of“Randomly”Flipping 2CoinsSobrietyCheckpointsNormalDistributionWorking rmalApproximationAccurateYou poll 100 people.What is the probability that exactly 65 report approval?Solution: Y Binomial(100, .7)
Suppose 70% approve the President . . .Math 1710Class 8V2BinomialTwo Ways of“Randomly”Flipping 2CoinsSobrietyCheckpointsNormalDistributionWorking rmalApproximationAccurateYou poll 100 people.What is the probability that exactly 65 report approval?Solution: Y Binomial(100, .7)P(Y 65) ?
Suppose 70% approve the President . . .Math 1710Class 8V2BinomialTwo Ways of“Randomly”Flipping 2CoinsSobrietyCheckpointsNormalDistributionWorking rmalApproximationAccurateYou poll 100 people.What is the probability that exactly 65 report approval?Solution: Y Binomial(100, .7) 100P(Y 65) .765 .335 .65
Suppose 70% approve the President . . .Math 1710Class 8V2BinomialTwo Ways of“Randomly”Flipping 2CoinsSobrietyCheckpointsNormalDistributionWorking rmalApproximationAccurateYou poll 100 people.What is the probability that exactly 65 report approval?Solution: Y Binomial(100, .7) 100P(Y 65) .765 .335 .65 100A calculator could help with.65
Calculating CombinationsMath 1710Class 8V2BinomialTwo Ways of“Randomly”Flipping 2CoinsSobrietyCheckpointsNormalDistributionWorking rmalApproximationAccurate 100A calculator could help with.65
Calculating CombinationsMath 1710Class 8V2BinomialTwo Ways of“Randomly”Flipping 2CoinsSobrietyCheckpointsNormalDistributionWorking rmalApproximationAccurate 100A calculator could help with.65TI-83,84 100 Math Prb nCr 65(Math is at the left of row 3.)TI-89 Math Probability nCr (100, 65)(Math is above the 5.)TI-30 100 nCr 65(nCr is above the 8 on my TI-30.)
Calculating CombinationsMath 1710Class 8V2BinomialTwo Ways of“Randomly”Flipping 2CoinsSobrietyCheckpointsNormalDistributionWorking rmalApproximationAccurate 100.65TI-83,84 100 Math Prb nCr 65(Math is at the left of row 3.)A calculator could help withTI-89 Math Probability nCr (100, 65)(Math is above the 5.)TI-30 100 nCr 65(nCr is above the 8 on my TI-30.)An answer like 1.095067153E 27 means 1.095 1027 and so 100P(Y 65) .765 .335 .04678.65
TediousMath 1710Class 8V2BinomialTwo Ways of“Randomly”Flipping 2CoinsSobrietyCheckpointsNormalDistributionWorking rmalApproximationAccurateNotice that calculating P(60 Y 65) by the above methodwould not be pleasant.
TediousMath 1710Class 8V2BinomialTwo Ways of“Randomly”Flipping 2CoinsSobrietyCheckpointsNormalDistributionWorking rmalApproximationAccurateNotice that calculating P(60 Y 65) by the above methodwould not be pleasant.We’ll see that an important technique called normalapproximation will get us quickly to that kind of answer.
TediousMath 1710Class 8V2BinomialTwo Ways of“Randomly”Flipping 2CoinsSobrietyCheckpointsNotice that calculating P(60 Y 65) by the above methodwould not be pleasant.We’ll see that an important technique called normalapproximation will get us quickly to that kind of answer.NormalDistributionWorking rmalApproximationAccurateTI-8x calculators have a binomialcdf function which can do this.Please don’t use that function to supply any homework orexam answers in this course.
X1 X2 vs. 2XMath 1710Class 8V2BinomialTwo Ways of“Randomly”Flipping 2CoinsSobrietyCheckpointsNormalDistributionWorking rmalApproximationAccurateTossing one fair coin is described by Bernoulli(.5):X01probability.5.5
X1 X2 vs. 2XMath 1710Class 8V2BinomialTwo Ways of“Randomly”Flipping 2CoinsTossing one fair coin is described by tionWorking rmalApproximationAccurateThe RV 2X ?probability.5.5
X1 X2 vs. 2XMath 1710Class 8V2BinomialTossing one fair coin is described by Bernoulli(.5):Two Ways of“Randomly”Flipping 2CoinsSobrietyCheckpointsNormalDistributionWorking bability.5.5The RV 2X :
X1 X2 vs. 2XMath 1710Class 8V2Tossing one fair coin is described by Bernoulli(.5):BinomialTwo Ways of“Randomly”Flipping 2CoinsSobrietyCheckpointsNormalDistributionWorking bability.5.5The RV 2X :If X1 and X2 are independent copies of X , then X1 X2 cancome out to 0,1, or 2.
X1 X2 vs. 2XMath 1710Class 8V2Tossing one fair coin is described by Bernoulli(.5):BinomialTwo Ways of“Randomly”Flipping lity.5.52X02probability.5.5The RV 2X :Working rmalApproximationAccurateX01The RV X1 X2 ?
X1 X2 vs. 2XMath 1710Class 8Tossing one fair coin is described by 5.5BinomialTwo Ways of“Randomly”Flipping 2CoinsThe RV 2X :SobrietyCheckpointsNormalDistributionWorking rmalApproximationAccurateThe RV X1 X2 :X1 X2012probability.25.5.25
How many heads in total?Math 1710Class 8V2BinomialTwo Ways of“Randomly”Flipping 2CoinsSobrietyCheckpointsNormalDistributionWorking rmalApproximationAccurateHow can these two possibilities come up in tossing 2 coins?
How many heads in total?Math 1710Class 8V2BinomialTwo Ways of“Randomly”Flipping 2CoinsSobrietyCheckpointsNormalDistributionWorking rmalApproximationAccurateHow can these two possibilities come up in tossing 2 coins?Method 1: Just toss them.
How many heads in total?Math 1710Class 8V2BinomialTwo Ways of“Randomly”Flipping 2CoinsSobrietyCheckpointsNormalDistributionWorking rmalApproximationAccurateHow can these two possibilities come up in tossing 2 coins?Method 1: Just toss them.This is X1 X2 .
How many heads in total?Math 1710Class 8V2BinomialTwo Ways of“Randomly”Flipping 2CoinsSobrietyCheckpointsNormalDistributionWorking rmalApproximationAccurateHow can these two possibilities come up in tossing 2 coins?Method 1: Just toss them.This is X1 X2 .Method 2: Toss one coin.Then turn the other coin over to the same result.
How many heads in total?Math 1710Class 8V2BinomialTwo Ways of“Randomly”Flipping 2CoinsHow can these two possibilities come up in tossing 2 coins?Method 1: Just toss them.SobrietyCheckpointsThis is X1 X2 .NormalDistributionMethod 2: Toss one coin.Then turn the other coin over to the same result.Working rmalApproximationAccurateNote the second coin is still random.
How many heads in total?Math 1710Class 8V2BinomialTwo Ways of“Randomly”Flipping 2CoinsHow can these two possibilities come up in tossing 2 coins?Method 1: Just toss them.SobrietyCheckpointsThis is X1 X2 .NormalDistributionMethod 2: Toss one coin.Then turn the other coin over to the same result.Working rmalApproximationAccurateThis is 2X .
Experimentally X1 X2 and 2XMath 1710Class 8V2BinomialTwo Ways of“Randomly”Flipping 2CoinsSobrietyCheckpointsNormalDistributionWorking rmalApproximationAccurateX1 X2012frequency?
Experimentally X1 X2 and 2XMath 1710Class 8V2BinomialTwo Ways of“Randomly”Flipping 2CoinsX1 X2012SobrietyCheckpointsNormalDistributionWorking rmalApproximationAccuratex̄,s?frequency?
Experimentally X1 X2 and 2XMath 1710Class 8V2BinomialTwo Ways of“Randomly”Flipping 2CoinsSobrietyCheckpointsNormalDistributionWorking rmalApproximationAccurateX1 X2012frequency?These x̄,s should be close to µ 1, σ resp.p2(.5)(.5) .707
Experimentally X1 X2 and 2XMath 1710Class 8V2BinomialTwo Ways of“Randomly”Flipping 2CoinsSobrietyCheckpointsNormalDistributionWorking rmalApproximationAccurate2X02frequency?
Experimentally X1 X2 and 2XMath 1710Class 8V2BinomialTwo Ways of“Randomly”Flipping king rmalApproximationAccuratex̄,s?frequency?
Experimentally X1 X2 and 2XMath 1710Class 8V2BinomialTwo Ways of“Randomly”Flipping ibutionWorking rmalApproximationAccurateThese x̄,s should be close to µ 1, σ 2(.5) 1 resp.
Sobriety CheckpointsMath 1710Class 8V2BinomialTwo Ways of“Randomly”Flipping 2CoinsSobrietyCheckpointsNormalDistributionWorking rmalApproximationAccurateOfficers ask questions, then maybe detain for a breathylyzertest.Suppose 12% of drivers nationally drink. (Inappropriately interms of driving.)Officers have right idea about drinking or not drinking about80% of the time.1)P(someone not drinking is detained for test)?2)P(being detained)?3)P(someone detained has been drinking)?4)P(someone released has not been drinking)?
A Continuous DistributionMath 1710Class 8V2BinomialTwo Ways of“Randomly”Flipping 2CoinsSobrietyCheckpointsNormalDistributionWorking rmalApproximationAccurateThere’s a normal distribution with any mean µ or σ 0.
A Continuous DistributionMath 1710Class 8V2BinomialTwo Ways of“Randomly”Flipping 2CoinsSobrietyCheckpointsNormalDistributionWorking rmalApproximationAccurateThere’s a normal distribution with any mean µ or σ 0.N(µ, σ)
A Continuous DistributionMath 1710Class 8V2BinomialTwo Ways of“Randomly”Flipping 2CoinsSobrietyCheckpointsNormalDistributionWorking rmalApproximationAccurateN(µ, σ)
A Continuous DistributionMath 1710Class 8V2N(µ, σ)BinomialTwo Ways of“Randomly”Flipping 2CoinsSobrietyCheckpointsNormalDistributionWorking rmalApproximationAccurateArea corresponds to probability.
A Continuous DistributionMath 1710Class 8V2BinomialN(µ, σ)Two Ways of“Randomly”Flipping 2CoinsSobrietyCheckpointsNormalDistributionWorking rmalApproximationAccurateThe entire area under the curve is 1.
A Continuous DistributionMath 1710Class 8V2N(µ, σ)BinomialTwo Ways of“Randomly”Flipping 2CoinsSobrietyCheckpointsNormalDistributionWorking rmalApproximationAccurateThe area between a and b is the probability of a value x fallingwithin that range.
The 68 95 99.7 RuleMath 1710Class 8V2BinomialTwo Ways of“Randomly”Flipping 2CoinsSobrietyCheckpointsNormalDistributionWorking rmalApproximationAccurate68% within 1 standard deviation of the mean.
The 68 95 99.7 RuleMath 1710Class 8V2BinomialTwo Ways of“Randomly”Flipping 2CoinsSobrietyCheckpointsNormalDistributionWorking rmalApproximationAccurate68% within 1 standard deviation of the mean.N(µ, σ)
The 68 95 99.7 RuleMath 1710Class 8V2BinomialTwo Ways of“Randomly”Flipping 2CoinsSobrietyCheckpointsNormalDistributionWorking rmalApproximationAccurate95% within 2 standard deviations of the mean.N(µ, σ)
The 68 95 99.7 RuleMath 1710Class 8V2BinomialTwo Ways of“Randomly”Flipping 2CoinsSobrietyCheckpointsNormalDistributionWorking rmalApproximationAccurate99.7% within 3 standard deviations of the mean.N(µ, σ)
The Standard Normal DistributionMath 1710Class 8V2BinomialTwo Ways of“Randomly”Flipping 2CoinsSobrietyCheckpointsNormalDistributionWorking rmalApproximationAccurateStandard normal is the case µ 0 and σ 1.
The Standard Normal DistributionMath 1710Class 8V2BinomialTwo Ways of“Randomly”Flipping 2CoinsSobrietyCheckpointsNormalDistributionWorking rmalApproximationAccurateStandard normal is the case µ 0 and σ 1.N(0,1)
The Standard Normal DistributionMath 1710Class 8V2BinomialTwo Ways of“Randomly”Flipping 2CoinsSobrietyCheckpointsNormalDistributionWorking rmalApproximationAccurateA general normal distribution:N(µ, σ)
The Standard Normal DistributionMath 1710Class 8N(µ, σ)V2BinomialTwo Ways of“Randomly”Flipping 2CoinsSobrietyCheckpointsNormalDistributionWorking rmalApproximationAccurateCan convert from a general N(µ, σ) to N(0, 1) via the Z-score.z x µσ
The Standard Normal DistributionMath 1710Class 8N(µ, σ)V2BinomialTwo Ways of“Randomly”Flipping 2CoinsSobrietyCheckpointsNormalDistributionWorking rmalApproximationAccuratex µσThe Z score is just the offset from the mean in standarddeviation units.z
The Standard Normal DistributionMath 1710Class 8V2BinomialTwo Ways of“Randomly”Flipping 2CoinsSobrietyCheckpointsNormalDistributionWorking rmalApproximationAccuratez x µσThis transformation preserves area and probability.
UsingTable ZMath 1710Class 8V2BinomialTwo Ways of“Randomly”Flipping 2CoinsSobrietyCheckpointsNormalDistributionWorking rmalApproximationAccurateN(0,1)
UsingTable ZMath 1710Class 8V2N(0,1)BinomialTwo Ways of“Randomly”Flipping 2CoinsSobrietyCheckpointsNormalDistributionWorking rmalApproximationAccurateTable Z gives us the area to the left on the standard normal.
UsingTable ZMath 1710Class 8V2N(0,1)BinomialTwo Ways of“Randomly”Flipping 2CoinsSobrietyCheckpointsNormalDistributionWorking rmalApproximationAccurateTable Z gives us the area to the left on the standard normal.i.e. P(Z z)
UsingTable ZMath 1710Class 8V2BinomialTwo Ways of“Randomly”Flipping 2CoinsSobrietyCheckpointsNormalDistributionWorking rmalApproximationAccurateTable Z gives us the area to the left on the standard normal.For example P(Z 1) .8413 since
UsingTable ZMath 1710Class 8V2BinomialTable Z gives us the area to the left on the standard normal.For example P(Z 1) .8413 sinceTwo Ways of“Randomly”Flipping 2CoinsSobrietyCheckpointsNormalDistributionWorking rmalApproximationAccurateAnd P(Z 1.16) .8770.
UsingTable ZMath 1710Class 8V2BinomialTwo Ways of“Randomly”Flipping 2CoinsSobrietyCheckpointsNormalDistributionWorking rmalApproximationAccurateTable Z gives us the area to the left on the standard normal.Note you use row 1.1 and column .06 for this!
UsingTable ZMath 1710Class 8V2BinomialTwo Ways of“Randomly”Flipping 2CoinsSobrietyCheckpointsNormalDistributionWorking rmalApproximationAccurateWe can also get the area under the curve between two values.
UsingTable ZMath 1710Class 8V2BinomialTwo Ways of“Randomly”Flipping 2CoinsSobrietyCheckpointsNormalDistributionWorking rmalApproximationAccurateFor example P( .67 Z 1) ?
UsingTable ZMath 1710Class 8V2BinomialTwo Ways of“Randomly”Flipping 2CoinsSobrietyCheckpointsNormalDistributionWorking rmalApproximationAccurateFor example P( .67 Z 1) ?N(0,1)
UsingTable ZMath 1710Class 8V2BinomialTwo Ways of“Randomly”Flipping 2CoinsSobrietyCheckpointsNormalDistributionWorking rmalApproximationAccurateTable Z tells us P( .67 Z 1) .8413 .2514 .5899.
UsingTable ZMath 1710Class 8V2BinomialTwo Ways of“Randomly”Flipping 2CoinsThe reason for the subtractionP( .67 Z 1) P(Z 1) P(Z .67) is:SobrietyCheckpointsNormalDistributionWorking rmalApproximationAccurate-
UsingTable ZMath 1710Class 8V2BinomialTwo Ways of“Randomly”Flipping 2CoinsThe reason for the subtractionP( .67 Z 1) P(Z 1) P(Z .67) is:SobrietyCheckpointsNormalDistributionWorking rmalApproximationAccurate
Mileage ExampleMath 1710Class 8V2BinomialTwo Ways of“Randomly”Flipping 2CoinsSobrietyCheckpointsNormalDistributionWorking rmalApproximationAccurateSuppose a normal model N(24 mpg, 6 mpg) describes fuelefficiency of cars in a region:
Mileage ExampleMath 1710Class 8V2BinomialTwo Ways of“Randomly”Flipping 2CoinsSobrietyCheckpointsNormalDistributionWorking rmalApproximationAccuratePercent of cars with mileage below 15?
Mileage ExampleMath 1710Class 8V2BinomialTwo Ways of“Randomly”Flipping 2CoinsSobrietyCheckpointsNormalDistributionWorking rmalApproximationAccurate% between 20 and 30?
Mileage ExampleMath 1710Class 8V2BinomialTwo Ways of“Randomly”Flipping 2CoinsSobrietyCheckpointsNormalDistributionWorking rmalApproximationAccurate% of cars above 40?
Mileage ExampleMath 1710Class 8V2BinomialTwo Ways of“Randomly”Flipping 2CoinsSobrietyCheckpointsNormalDistributionWorking rmalApproximationAccurateWorst 20% of cars?
Mileage ExampleMath 1710Class 8V2BinomialTwo Ways of“Randomly”Flipping 2CoinsSobrietyCheckpointsNormalDistrib
Math 1710 Class 8 V2 Binomial Two Ways of \Randomly" Flipping 2 Coins Sobriety Checkpoints Normal Distribution Working With Normal Distributions Normal Approximation Making Normal Approximation Accurate
1 Launch the Dell Local Printer Settings Utility by clicking Start Programs Dell Printers Dell Laser Printer 1710 Dell Local Printer Settings Utility. 2 Select your Dell Laser Printer 1710. 3 Click OK. 4 In the left column, click Paper. 5 In the right column, go to the Tray 1 section
Quantum machine learning for quantum anomaly detection NANA LIU CQT AND SUTD, SINGAPORE ARXIV:1710.07405 TUESDAY 7TH . Chandola et al, Anomaly detection: a survey, ACM computing surveys, 2009 arXiv:1710.07405. Types of anomalies Supervised Unsupervised Semi-supervised Point Contextual Collective arXiv:1710.07405. Types of anomalies Supervised .
Math 5/4, Math 6/5, Math 7/6, Math 8/7, and Algebra 1/2 Math 5/4, Math 6/5, Math 7/6, Math 8/7, and Algebra ½ form a series of courses to move students from primary grades to algebra. Each course contains a series of daily lessons covering all areas of general math. Each lesson
There will be an optional (to do) lab on Chi Square the day of prelim 2, but most of that recitation will be devoted to review. Consequently tomorrow’s lab includes elements of chapters 23, 24, and 25. Math 1710 Class 32 V2cu Tomorrow’s Lab Coffee Machine t-CI’s and HT’s t-Distributions Shaft Diameters A Sample Size Problem An Indep
MATH 110 College Algebra MATH 100 prepares students for MATH 103, and MATH 103 prepares students for MATH 110. To fulfil undergraduate General Education Core requirements, students must successfully complete either MATH 103 or the higher level MATH 110. Some academic programs, such as the BS in Business Administration, require MATH 110.
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Geburtstagskolloquium Reinhard Krause-Rehberg Andreas Wagner I Institute of Radiation Physics I www.hzdr.de Member of the Helmholtz AssociationPage Positrons slow down to thermal energies in 3-10 ps. After diffusing inside the matter positrons are trapped in vacancies or defects. Kinetics results in trapping rates about
