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Prob & Stat: Islamic Approach
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L1 Random Samples
1 Models Versus Reality
ISIA L01 C1 S1
ISIA L01 C1 S2
ISIA L01 C1 S3
ISIA L1 C1 S4
ISIA L1 C1 S5
ISIA L1 C1 S6
2 Populations
ISIA P&S L1 C2 S1
ISIA P&S L1 C2 S2
ISIA P&S L1 C2 S3
3 Random Draws
P&S L1 C3 S2
P&SIA L1 C3 S1
P&SIA L1 C3 S3
P&SIA L1 C3 S4
P&SIA L1 C3 S5
4 Simple Random Samples
P&SIA L1 C4 S2
P&SIA L1 C4 S3
P&SIA L1 C4 S4
P&SIA L1 C4 S5
P&SIA L1 C4 S6
P&SIA L1C4 S1
5 Law of Large Numbers
P&S L1 C5 S1
P&S L1 C5 S2
P&S L1 C5 S3
P&S L1 C5 S4
P&S L1 C5 S5
L2 Bernoulli Sequences
1 Bernoulli RV's
P&S L2 C1 S1
P&S L2 C1 S2
P&S L2 C1 S3
P&S L2 C1 S4
P&S L2 C1 S5
2 Laws of Probability
P&S L2 C2 S1
P&S L2 C2 S2
P&S L2 C2 S3
P&S L2 C2 S4
P&S L2 C2 S5
3 The Multiplication Law
L2 C3 S1
L2 C3 S2
L2 C3 S3
L2 C3 S4
L2 C3 S5
answer
4 Bernoulli Probabilities
L2 C4 S1
L2 C4 S2
L2 C4 S3
L2 C4 S4
Solutions
L2 C4 S5
5 Pre- & Post-Experiment
L2 C4 S4
L2 C5 S1
L2 C5 S2
L2 C5 S3
L3 Discrete Random Variables
1: Uniform RV
1 Equal Chances for All
2: Real World Representations
3 Testing Randomness
4 Applications
5 Examples
2: Binomial RV's
S1 Complex Random Draw
S2 Binomial as Sum of Bernoulli's
S3 Binomial Distribution
S4 Real World Examples: Voters
S5 RWE(2): Treatments
3: Binomial Calculations
1 EXCEL Calculations
2 Binomial Problems
Answers
3 Deviations Reduce in Large Samples
4 Conditions for Binomial
5 Counterexamples
4 General Discrete Finite RV's
S1 EXCEL RV's
S2 Multinomial
S3 Multinomial Exercises
S4 Simulations
S5 Convergence of Observed Proportions to Probabilities
5 Poisson RV's
S1 Definition
S2 Event Counts
S3 Poisson Calculations
S4 Counterexamples
S5 Binomial Approximation
S6 More Examples
S7 Exercises
L4 Continuous Random Variables
1: Densities & Distributions (CDF)
L4 C1 S1 Definition and Intuition
S2 Cumulative Distribution Functions
S3 Limits at infinities
S4 Monotonicity
S5 Continuity from Right; Limits from Left
S6 Equivalence between CDF and RV
2 Uniform RV's
S1 Uniform Density
S2 Three Uniforms
S3 Uniform CDF
S4 Approximation by Discrete Uniforms
S5 Real World Examples
3: Gamma RV's
S1 Gamma Function
L4C3S1 Properties of Gamma Functions
S2 Gamma RV
S3 Gamma Densities
S4 Real World Examples
S5 Understanding Models
4: Normal RV's
S1 Normal Density
S2 Elementary Properties
S3 Generating Normal RV's
Exponential RV
S4 Verifying Elementary Properties
S5 Identifying Distributions from Data
5 Change of Variables
S1 C-of-V Formula
S2 Location Scale Families
S3 General Normal Distributions
S4 Square of Normal
S5 The Probability Integral Transform
L5 Expectations, Moments, & MGF
1 Expectation
S1 The Discrete Case
S2 Binomial
S3 Poisson
S4 Gamma
S5 Normal
2 Moments
S1 Defining Moments
S2 Raw and Central Moments
S3 The Normal Distribution
S4 Standardization
S5 Skewness
S6 Kurtosis
3 Moment Generating Functions
S1 Definition of MGF
S2 MGF of Normal
S3 MGF of Gamma
S4 MGF of Binomial
S5 MGF of Poisson
4 Sums of Independent RV's
S1 MGF of Sums
S2 Sums of Normals
S3 Sums of Gammas
S4 Sums of Binomials
S5 Sums of Poisson's
5 Law of Large Numbers
S1 LLN Explanation & Intuition
S2 LLN via Chebyshev Inequality
S3 Illustration of Use
S4 Swamping vs Balancing
S5 The Regression Effect
L6 Central Limit Theorem
1 Sums Are Normal
S1 Cumulant Generating Function
S2 Cumulants and Moments
S3 Properties of Cumulants
S4 Central Limit Theorem
S5 Generalizations of CLT
2 Binomial CLT
S1 Binomial CLT
S2 Binomial CLT via Bernoulli Sums
S3 De-Moivre Laplace CLT
S4 Accuracy of Normal Approximation
S5 Application
3 Chi Square CLT
S1 Definition of Chi-Square
S2 Sums of Squared Normals
S3 Cochran's Theorem
S4 Cumulants and CLT
S5 Application
4 Errors in Survey Samples
S1 Estimates of Sampling Error
S2 Probability and Confidence
S3 Sample Size for desired Confidence Interval
S4 Specifying Confidence Levels and Interval Sizes
S5 Pilot Studies
5 Simulations
S1 Estimating Probabilities
S2 Mean of a Lognormal
S3 Moments & CLT
S4 Analytical & Numerical Approach
S5 Estimating Distribution
L7 Transition: Multivariate
1 Distributions & IID RVs
2 Methodology
3 Bivariate RVs & Dist
L8: Multivariate
OUTLINE
Prob & Stat: Islamic Approach
S3 The Normal Distribution
L5 C2 S3.docx
Properties of Normal Distribution
— Computation of Moments is done by integration by parts.
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