mathematics and statistics boot camp ii david siroky duke university

Mathematics and Mathematics and Statistics Boot Camp IIStatistics Boot Camp II

David SirokyDavid Siroky

Duke UniversityDuke University

AgendaAgenda The Language of MathematicsThe Language of Mathematics Algebra ReviewAlgebra Review ExponentsExponents LogarithmsLogarithms Problem Set ReviewProblem Set Review Probability ReviewProbability Review

Will most likely end here…Will most likely end here… Limits and Continuity Limits and Continuity Differential Calculus (Tangents, Differentiation, Differential Calculus (Tangents, Differentiation,

Extrema)Extrema) Integral Calculus (Integration Rules, Definite Integral Calculus (Integration Rules, Definite

Integrals)Integrals) Matrix Algebra (Determinants, Inverses, Matrix Algebra (Determinants, Inverses,

Eigenvalues and Eigenvectors)Eigenvalues and Eigenvectors)

The Language of The Language of Mathematics 0Mathematics 0

+ + -- XX == >> << | x || x |

AdditionAddition SubtractionSubtraction MultiplicationMultiplication DivisionDivision EqualityEquality Greater thanGreater than Less thanLess than Absolute value of xAbsolute value of x Square RootSquare Root

The Language of The Language of Mathematics IMathematics I

||

- There exists- There exists - For all- For all - Therefore- Therefore - Implies- Implies - Such that, Given- Such that, Given - Change- Change - Element (- Element ())

The Language of The Language of Mathematics IIMathematics II

- Exactly equal- Exactly equal - Roughly equal- Roughly equal - Not equal- Not equal - Summation- Summation - Product- Product - Partial Derivative- Partial Derivative - Integral- Integral

The Language of Mathematics IIIThe Language of Mathematics III

ln ln loglogbb

22

Natural Log Natural Log Log to Base Log to Base bb Mu = meanMu = mean Sigma = Std Dev.Sigma = Std Dev. Sigma sq. = Var.Sigma sq. = Var. UnionUnion IntersectionIntersection

Quick ReviewQuick Review

[e.g., [e.g., XXii]] [e.g., [e.g., y/ y/ x]x]| [e.g., Pr (Y | X)]| [e.g., Pr (Y | X)] [e.g., A U B][e.g., A U B] [e.g., [e.g., xx]]

Algebra Review – Algebra Review – 10 Commandments of 10 Commandments of

ExponentsExponents 1. A1. Am x m x AAn = n = AAm + nm + n

2. (A2. (Amm))n = n = A A m x nm x n

3. (A 3. (A xx B) B)n = n = AAn +n +BB n n

4. (A/B)4. (A/B)n n =(A=(Ann/B/Bnn) )

BB00 5. (1/A5. (1/Ann) = A) = A-n-n

6. (A6. (Amm/A/Ann) ) == A Am – n m – n

==1/A1/An - mn - m

7. A 7. A ½ ½ == AA 8. A 8. A 1/n 1/n == n nAA 9. A 9. A m/n m/n == (A (A 1/n1/n))m m

== (A (A mm))1/n 1/n = = nnAAmm

10. A10. A00 = 1 b/c A = 1 b/c A0 0

= A= A n – n n – n

= A= Ann/A/An n = 1= 1

Algebra Review – Algebra Review – Some Examples of ExponentsSome Examples of Exponents

Definition: 6Definition: 633

= 6 x 6 x 6 = 216= 6 x 6 x 6 = 216 [#2] (5[#2] (522))2 2

= 5 = 5 2 x 22 x 2 = 5 = 5 44 = 625 = 625 [#3] (3 x 4)[#3] (3 x 4)22

= 3 = 3 22 x 4 x 4 2 2 = 9 x 16 = 144= 9 x 16 = 144 [#4] (1/16)[#4] (1/16)1/41/4

= (1= (11/41/4/16/161/41/4) = [# 9] ) = [# 9] (1(11/41/4/(2/(244))1/41/4) = (1) = (11/41/4/2/24/44/4) = ½ ) = ½

Examples from Problem Set 1.Examples from Problem Set 1.

Problem Set Exponent Problem Set Exponent ExamplesExamples

Problem Set ExponentsProblem Set Exponents

Problem Set ExponentsProblem Set Exponents

Problem Set ExponentsProblem Set Exponents

Algebra Review –Algebra Review –Some of the Rules for Some of the Rules for

LogarithmsLogarithms Log (A x B)Log (A x B) Log (A/B)Log (A/B) Log (ALog (Ann)) Ln eLn exx

eeln xln x

= log (A) + log (B)= log (A) + log (B) = log (A) – log (B)= log (A) – log (B) = n log A= n log A = x= x = x= x

An Example of the First Rule for An Example of the First Rule for LogarithmsLogarithms

log 100 + log 400log 100 + log 400 [#1] Log 40,000 = [#1] Log 40,000 = log (10,000 x 4) = log (10,000 x 4) = log 10,000 + log 4 log 10,000 + log 4 =log 10=log 1044 + log 4 = + log 4 = 4 + .6 4 + .6 4.6 4.6

[#1] Log 10[#1] Log 1022 + log + log (4 x 10(4 x 1022) = log 10) = log 102 2

+ log 4 + log 10+ log 4 + log 1022 = = 2 + log 4 + 2 2 + log 4 + 2 4.6 4.6

Problem Set 1 Logarithms: 4aProblem Set 1 Logarithms: 4a

Problem Set 1 Logarithms: 4bProblem Set 1 Logarithms: 4b

Problem Set 1 Logarithms: 4cProblem Set 1 Logarithms: 4c

Last Questions from Problem Set Last Questions from Problem Set 11

Other Problem Set Topics: Other Problem Set Topics: GraphingGraphing

ProbabilityProbability TheoryTheory

Probabilities and OutcomesProbabilities and OutcomesSample space: set of all possible Sample space: set of all possible

outcomesoutcomesEvent: subset of sample spaceEvent: subset of sample spaceRandom VariablesRandom VariablesProbability DistributionProbability DistributionAn ExampleAn Example

Probability Distribution of Probability Distribution of Discrete Random variableDiscrete Random variable

List of all possible values of the List of all possible values of the variablevariable

And the probability that each will And the probability that each will occur.occur.

Must sum to 1Must sum to 1

For exampleFor example

Let M be the number of times your Let M be the number of times your computer crashes using Windows.computer crashes using Windows.

Probability that M=0 is Pr (M=0)Probability that M=0 is Pr (M=0)Probability that M=1 is Pr (M=1) etc.Probability that M=1 is Pr (M=1) etc.

Windows Crash TableWindows Crash Table

CrashesCrashes 00 11 22 33 44

Probability Probability DistributioDistributionn

.8.8 .1.1 .06.06 .03.03 .01.01

Cumulative Cumulative DistributioDistributionn

.8.8 .9.9 .96.96 .99.99 1.001.00

Windows Crash Probability Windows Crash Probability Distribution: A HistogramDistribution: A Histogram

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 1 2 3 4

Probability Distribution

Cumulative Probability Cumulative Probability Distribution (CDF)Distribution (CDF)

0

0.2

0.4

0.6

0.8

1

0 1 2 3 4

Cumulative Probability Distribution

Bernoulli DistributionBernoulli Distribution

For Discrete Dichotomous VariablesFor Discrete Dichotomous VariablesLet G be the gender of next person Let G be the gender of next person

you meet, where G= 0 if male and G you meet, where G= 0 if male and G = 1 if female. = 1 if female.

The outcomes and probabilities are:The outcomes and probabilities are:

G = 1 with probability G = 1 with probability pp

G = 0 with probability G = 0 with probability 1-p1-p

Values of InterestValues of Interest

Probabilities, First DifferencesProbabilities, First DifferencesExpected Value of a Random Expected Value of a Random

VariableVariable

E (Y) is the long run average of E (Y) is the long run average of many repeated trials or occurrences.many repeated trials or occurrences.

Also called the expectation of Y Also called the expectation of Y

Also called the mean of Y, e.g., (Also called the mean of Y, e.g., (y y ) = ) = MuMuYY

For ExampleFor Example

You loan a friend $100 at 10 % interest.You loan a friend $100 at 10 % interest. If repaid in full you get $110If repaid in full you get $110But there is a risk of 1% that your But there is a risk of 1% that your

friend will default and you get nothing. friend will default and you get nothing.

So the amount you get is a random So the amount you get is a random variable that equal 110 with probability variable that equal 110 with probability .99 and 0 with probability .01..99 and 0 with probability .01.

On average, On average, , what you get = 110 , what you get = 110 x .99 + 0 x .01 = 108.9x .99 + 0 x .01 = 108.9

Back to Windows CrashesBack to Windows Crashes

E (M) = 0 (.8) + 1 (.1) + 2 (.06) + 3 E (M) = 0 (.8) + 1 (.1) + 2 (.06) + 3 (.03) + 4 (.01) = .35(.03) + 4 (.01) = .35

This is the expected number of This is the expected number of crashes while working on your crashes while working on your Windows OS. Windows OS.

Of course the actual number is an Of course the actual number is an integer integer

Expectation for Bernoulli RVExpectation for Bernoulli RV

E (G) = 1 x p + 0 (1-p) = pE (G) = 1 x p + 0 (1-p) = p

Or the probability that the value Or the probability that the value assumed in 1 (female).assumed in 1 (female).

Expectation of Continuous RVExpectation of Continuous RV

E (Y) = yE (Y) = y11pp11 + y + y22pp22 + … + y + … + ykk p pkk

KK

== y yiippii

i=1i=1

Other topicsOther topics

Variance, St. Dev., MomentsVariance, St. Dev., MomentsCondition ExpectationCondition Expectation IndependenceIndependenceStandard NormalStandard NormalLaw of Large NumbersLaw of Large NumbersCentral Limit TheoremCentral Limit Theorem

ReviewReview

OutcomesOutcomes ProbabilityProbability Sample SpaceSample Space EventEvent Discrete RVDiscrete RV Continuous RVContinuous RV Bernoulli RVBernoulli RV CDFCDF

Expected ValueExpected Value MomentsMoments Conditional Conditional

ExpectationExpectation Law of Iterated Law of Iterated

ExpectationsExpectations Law of Large Law of Large

NumbersNumbers