M1S Probability And Statistics I
| Chapter 1 Sample Spaces and Events | Chapter 4 Combinatorics and Probability | Chapter 7 Transformations |
| Chapter 2 Probability | Chapter 5 Discrete random variables | Chapter 8 Generalized Expectations |
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Chapter 3 Conditional Probability |
Chapter 6 Continuous random variables | Chapter 9 Joint Distributions |
CHAPTER 1 SAMPLE SPACES AND EVENTS Commentary
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CONTENT |
KEY DEFINITIONS |
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| Introduction | The motivation for the use of probability
theory; scientific background. |
Uncertainty | |
| 1.1 | Representing Uncertainty | The Mathematical Framework for Uncertainty
Calculations |
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| 1.2 | Manipulating Collections of Sample Outcomes | Set Theory basis for probability calculations | Sample space (Countable/Uncountable) Sample outcome/Actual outcome Events Certain Event/Impossible Event |
| 1.3 | Operations of Set Theory | Manipulations of Sets | Complement/Union/Intersection operators Mutually exclusive events Exhaustive events Venn Diagrams Elementary Results/De Morgan's Laws Associativity/Distributivity Simplification Networks Partitions |
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CHAPTER 2 PROBABILITY: DEFINITIONS, INTERPRETATIONS, BASIC LAWS Commentary
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CONTENT |
KEY DEFINITIONS |
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| 2.1 | The Meaning of Probability | Probability as a Set Function Three Interpretations Mathematical Rules |
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| 2.2 | Mathematical Rules of Probability | Axiomatic manipulation of uncertainty | The Probability Axioms Extensions |
| 2.3 | Corollaries to the Axioms | Results derivable from the three probability axioms | Probability of a complement event The General Addition Rule for two events/n events |
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CHAPTER 3 CONDITIONAL PROBABILITY Commentary
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CONTENT |
KEY DEFINITIONS |
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| Conditioning on new information | The interpretation and role of conditional probability | Conditional Probability Proof that the conditional probability obeys the probability axioms Calculations for simple examples (including the Medical Screening Example) Independence The Chain/General Multiplication Rule Probability Trees |
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| 3.1 | The Theorem of Total Probability | The First Probability Theorem | The statement and proof of the theorem (simple and general cases) Uses of the theorem |
| 3.2 | Bayes Theorem | The Second Probability Theorem | The statement and proof of the theorem (simple and general cases) Uses of the theorem: medical screening example; false positives/negatives. Odds |
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CHAPTER 4 COMBINATORICS AND PROBABILITY Commentary
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CONTENT |
KEY DEFINITIONS |
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| Introduction | Probability calculations in the equally likely outcomes case | Classical probability definition | |
| 4.1 | Counting Operations | Elementary counting techniques: combinatorial terminology | The Multiplication Principle Factorials Sampling from a Finite Population Sampling with/without replacement Ordered/Unordered samples Distinguishable/Undistinguishable items Permutation/Combination Generating functions Binary sequence representations
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| 4.2 | Combinatorial Identities | Some results for combinatorial manipulation |
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| 4.3 | The Hypergeometric Formula | Calculations for Finite Type I/Type II populations (URN MODELS) | The hypergeometric formula; context, interpretation, uses. |
| 4.4 | Combinatorial Partitioning | Population partitioning into subsets | The partition/multinomial formula Simple examples (e.g. Poker Hands) |
| 4.5 | Occupancy Problems | Allocation of items to cells (in the distinguishable and indistinguishable cases) | Examples: balls to cells with no cell empty, the Birthday problem etc. |
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CHAPTER 5 DISCRETE RANDOM VARIABLES AND DISTRIBUTIONS Commentary
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CONTENT |
KEY DEFINITIONS |
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| 5.1 | Random variables | A map from general sample space to real numbers | Random variable Range of random variable The Discrete Random Variable |
| 5.2 | The Probability Mass Function | The function to describe how probability is
distributed; functionals of the pmf |
Definition of the pmf Properties Examples and types of calculation Construction of pmfs Generating Functions The probability generating function (PGF); definition, uses, calculations. Expectation; definition and interpretation. |
| 5.3 | The Cumulative Distribution Function | Definition of the cdf Properties Relation to pmf Uses.
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| 5.4 | The Bernoulli Distribution | Definition (pmf) Experimental context (single 0/1 trial) PGF Expectation
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| 5.5 | The Binomial Distribution | Definition (pmf) Experimental context (n repeated 0/1 trials, count number of 1s) PGF Expectation Limiting case as n gets large (via pgf/pmf) |
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| 5.6 | The Poisson Distribution | Definition (pmf) Experimental context (limiting case of binomial/Poisson process) PGF Expectation Uses/Examples |
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| 5.7 | The Geometric Distribution | Definition (pmf/cdf) Experimental context (repeated 0/1 trials until first success - a waiting time distribution) PGF Expectation |
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| 5.8 | The Negative Binomial Distribution | Definition (pmf) Experimental context (repeated 0/1 trials until nth success) PGF Expectation Alternative Representation |
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| 5.9 | The Hypergeometric Distribution | Definition Experimental context (sampling from a finite population of Type I/Type II objects) |
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| 5.10 | The Probability Generating Function | A generating function for probabilities | Definition Uses Key result for sums of independent random variables Expectation calculation via the pgf Calculations for standard distributions |
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CHAPTER 6 CONTINUOUS RANDOM VARIABLES Commentary
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CONTENT |
KEY DEFINITIONS |
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| Uncountable sample spaces with continuous cdfs | |||
| 6.1 | Continuous Random variables | Definition | The uncountable space case |
| 6.2 | The Continuous Cumulative Distribution Function | Definition of the continuous cdf Properties |
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| 6.3 | The Probability Density Function | Definition of the pdf as the derivative of
the continuous cdf Properties Uses. Construction of pdfs Expectation for Continuous Variables
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| 6.4 | The Uniform Distribution | Definition (pdf) Experimental context (equally likely outcomes) Expectation
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| 6.5 | The Exponential Distribution | Definition (pdf/cdf) Experimental context (continuous waiting time distribution) Expectation Interpretation Connection to the Poisson Process The Lack of Memory Property
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| 6.6 | The Gamma Distribution | Construction The Gamma Function and its properties Definition (pdf) Experimental context Expectation Generating Function Uses/Examples Scale Transformations Special Case: The Chi-squared distribution |
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| 6.7 | The Normal Distribution | Construction: Standard case Definition (pdf) Location/scale transformation Generating Function Expectation |
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CHAPTER 7 TRANSFORMATIONS OF RANDOM VARIABLES Commentary
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CONTENT |
KEY DEFINITIONS |
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| General Approach to the Transformation of Random Variables | Discrete/Continuous case "First principles" calculation 1-1 transformations: the Jacobian Non 1-1 Transformations |
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CHAPTER 8 GENERALIZED EXPECTATIONS Commentary
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CONTENT |
KEY DEFINITIONS |
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| Expectations of a General Function of a random variable | Definition/application Properties: linearity Special cases: moments, central moments (variance) The PGF as an expectation The Moment Generating Function (MGF) and its uses (identification/moment calculation). Relationship between MGF and PGF The Central Limit Theorem |
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CHAPTER 9 JOINT DISTRIBUTIONS Commentary
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CONTENT |
KEY DEFINITIONS |
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| Probability distributions for vectors of random variables | Introduction via a 2-D discrete model Joint pmf/pdf Marginal pmf/pdf Conditional pmf/pdf
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