Mathematics and Statistics Boot Camp II David Siroky Duke University

Agenda The Language of Mathematics The Language of Mathematics Algebra Review Algebra Review Exponents Exponents Logarithms Logarithms Problem Set Review Problem Set Review Probability Review Probability Review Will most likely end here…  Will most likely end here…  Limits and Continuity Limits and Continuity Differential Calculus (Tangents, Differentiation, Extrema) Differential Calculus (Tangents, Differentiation, Extrema) Integral Calculus (Integration Rules, Definite Integrals) Integral Calculus (Integration Rules, Definite Integrals) Matrix Algebra (Determinants, Inverses, Eigenvalues and Eigenvectors) Matrix Algebra (Determinants, Inverses, Eigenvalues and Eigenvectors)

The Language of Mathematics X  = > < | x | | x |  Addition Addition Subtraction Subtraction Multiplication Multiplication Division Division Equality Equality Greater than Greater than Less than Less than Absolute value of x Absolute value of x Square Root Square Root

The Language of Mathematics 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 Mathematics 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 III ln ln log b log b    2  2   Natural Log Natural Log Log to Base b Log to Base b Mu = mean Mu = mean Sigma = Std Dev. Sigma = Std Dev. Sigma sq. = Var. Sigma sq. = Var. Union Union Intersection Intersection

Quick Review  [e.g.,  X i ]  [e.g.,  X i ]  [e.g.,  y/  x]  [e.g.,  y/  x] | [e.g., Pr (Y | X)] | [e.g., Pr (Y | X)]  [e.g., A U B]  [e.g., A U B]  [e.g.,  x ]  [e.g.,  x ]

Algebra Review – 10 Commandments of Exponents 1. A m x A n = A m + n 1. A m x A n = A m + n 2. (A m ) n = A m x n 2. (A m ) n = A m x n 3. (A x B) n = A n + B n 3. (A x B) n = A n + B n 4. (A/B) n =(A n /B n )  B  0 4. (A/B) n =(A n /B n )  B  0 5. (1/A n ) = A -n 5. (1/A n ) = A -n 6. (A m /A n ) = A m – n 6. (A m /A n ) = A m – n =1/A n - m =1/A n - m 7. A ½ =  A 7. A ½ =  A 8. A 1/n = n  A 8. A 1/n = n  A 9. A m/n = (A 1/n ) m 9. A m/n = (A 1/n ) m = (A m ) 1/n = n  A m = (A m ) 1/n = n  A m 10. A 0 = 1 b/c A A 0 = 1 b/c A 0 = A n – n = A n – n = A n /A n = 1 = A n /A n = 1

Algebra Review – Some Examples of Exponents Definition: 6 3 Definition: 6 3 = 6 x 6 x 6 = 216 = 6 x 6 x 6 = 216 [#2] (5 2 ) 2 [#2] (5 2 ) 2 = 5 2 x 2 = 5 4 = 625 = 5 2 x 2 = 5 4 = 625 [#3] (3 x 4) 2 [#3] (3 x 4) 2 = 3 2 x 4 2 = 9 x 16 = 144 = 3 2 x 4 2 = 9 x 16 = 144 [#4] (1/16) 1/4 [#4] (1/16) 1/4 = (1 1/4 /16 1/4 ) = [# 9] (1 1/4 /(2 4 ) 1/4 ) = (1 1/4 /2 4/4 ) = ½ = (1 1/4 /16 1/4 ) = [# 9] (1 1/4 /(2 4 ) 1/4 ) = (1 1/4 /2 4/4 ) = ½ Examples from Problem Set 1. Examples from Problem Set 1.

Problem Set Exponent Examples

Problem Set Exponents

Algebra Review – Some of the Rules for Logarithms Log (A x B) Log (A x B) Log (A/B) Log (A/B) Log (A n ) Log (A n ) Ln e x Ln e x e ln x e ln 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

An Example of the First Rule for Logarithms log log 400 log log 400 [#1] Log 40,000 = log (10,000 x 4) = log 10,000 + log 4 =log log 4 =  4.6 [#1] Log 40,000 = log (10,000 x 4) = log 10,000 + log 4 =log log 4 =  4.6 [#1] Log log (4 x 10 2 ) = log log 4 + log 10 2 = 2 + log  4.6 [#1] Log log (4 x 10 2 ) = log log 4 + log 10 2 = 2 + log  4.6

Problem Set 1 Logarithms: 4a

Problem Set 1 Logarithms: 4b

Problem Set 1 Logarithms: 4c

Last Questions from Problem Set 1

Other Problem Set Topics: Graphing

ProbabilityTheory Probabilities and Outcomes Probabilities and Outcomes Sample space: set of all possible outcomes Sample space: set of all possible outcomes Event: subset of sample space Event: subset of sample space Random Variables Random Variables Probability Distribution Probability Distribution An Example An Example

Probability Distribution of Discrete Random variable List of all possible values of the variable List of all possible values of the variable And the probability that each will occur. And the probability that each will occur. Must sum to 1 Must sum to 1

For example Let M be the number of times your computer crashes using Windows. Let M be the number of times your 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 Table Crashes01234 Probability Distribution Cumulative Distribution

Windows Crash Probability Distribution: A Histogram

Cumulative Probability Distribution (CDF)

Bernoulli Distribution For Discrete Dichotomous Variables For Discrete Dichotomous Variables Let G be the gender of next person you meet, where G= 0 if male and G = 1 if female. Let G be the gender of next person you meet, where G= 0 if male and G = 1 if female. The outcomes and probabilities are: The outcomes and probabilities are: G = 1 with probability p G = 0 with probability 1-p

Values of Interest Probabilities, First Differences Probabilities, First Differences Expected Value of a Random Variable Expected Value of a Random Variable E (Y) is the long run average of many repeated trials or occurrences. Also called the expectation of Y Also called the mean of Y, e.g., (  y ) = Mu Y

For Example You loan a friend $100 at 10 % interest. You loan a friend $100 at 10 % interest. If repaid in full you get $110 If repaid in full you get $110 But there is a risk of 1% that your friend will default and you get nothing.  But there is a risk of 1% that your friend will default and you get nothing.  So the amount you get is a random variable that equal 110 with probability.99 and 0 with probability.01. So the amount you get is a random variable that equal 110 with probability.99 and 0 with probability.01. On average, , what you get = 110 x x.01 = On average, , what you get = 110 x x.01 = 108.9

Back to Windows Crashes E (M) = 0 (.8) + 1 (.1) + 2 (.06) + 3 (.03) + 4 (.01) =.35 E (M) = 0 (.8) + 1 (.1) + 2 (.06) + 3 (.03) + 4 (.01) =.35 This is the expected number of crashes while working on your Windows OS. This is the expected number of crashes while working on your Windows OS. Of course the actual number is an integer Of course the actual number is an integer

Expectation for Bernoulli RV E (G) = 1 x p + 0 (1-p) = p E (G) = 1 x p + 0 (1-p) = p Or the probability that the value assumed in 1 (female). Or the probability that the value assumed in 1 (female).

Expectation of Continuous RV E (Y) = y 1 p 1 + y 2 p 2 + … + y k p k E (Y) = y 1 p 1 + y 2 p 2 + … + y k p k K =  y i p i =  y i p i i=1 i=1

Other topics Variance, St. Dev., Moments Variance, St. Dev., Moments Condition Expectation Condition Expectation Independence Independence Standard Normal Standard Normal Law of Large Numbers Law of Large Numbers Central Limit Theorem Central Limit Theorem

Review Outcomes Outcomes Probability Probability Sample Space Sample Space Event Event Discrete RV Discrete RV Continuous RV Continuous RV Bernoulli RV Bernoulli RV CDF CDF Expected Value Expected Value Moments Moments Conditional Expectation Conditional Expectation Law of Iterated Expectations Law of Iterated Expectations Law of Large Numbers Law of Large Numbers