UNIT 4 MATHEMATICAL METHODS

Transcription

1 UNIT 4 MATHEMATICAL METHODS PROBABILITY Section 1: Introductory Probability Basic Probability Facts Probabilities of Simple Events Overview of Set Language Venn Diagrams Probabilities of Compound Events Choices of Events The Addition Rule Combinations of Events The Multiplication Rule Dependent Events Conditional Probabilities Independent Events The Union and Intersection for Different Configurations of Two Events Mutually Exclusive vs Independency of Events Probability Diagrams Tree Diagrams Venn Diagrams Karnaugh Maps Section 2: Counting Techniques Factorials Permutations Combinations Section 3: Random Variables and their Distributions Discrete Random Variables Continuous Random Variables Probability Distributions and Functions Discrete Probability Distributions Continuous Probability Distributions The Cumulative Density Function Parameters Used to Describe Probability Functions Measures of Central Tendency The Mean Value (Expected Value) The Median The Mode

2 Measures of Spread The Range The Interquartile Range The Variance and Standard Deviation Shapes of Probability Distributions Section 4: General Discrete Probability Distributions Characteristics of Discrete Probability Distributions Finding Probabilities The Expected Value or Mean Properties of E(X) The Expected Value of a Function The Variance The Variance of Functions The Standard Deviation Two Sigma Property Section 5: The Binomial Distribution Properties of Bernoulli Sequences Using CAS Technology to Find Probabilities in Binomial Distributions The Mean and Standard Deviation of Binomial Distributions Calculating the Trial Size Given a Probability The Binomial Probability Distribution Graph Section 6: Select VCAA Questions Short Answer Questions Multiple Choice Questions Extended Response Questions Section 7: The Continuous Probability Distribution Calculating Probabilities of Simple PDFs Probabilities Involving Complex PDFs The Mean of Simple PDFs The Mean of Complex PDFs The Mean of Functions in Terms of X The Median of Simple PDFs The Median of Complex PDFs The Mode The Variance and Standard Deviation of Simple PDFs The Variance and Standard Deviation of Complex PDFs The Variance of Functions in Terms of X Sums of Independent Variables

3 The Range Percentiles and Quantiles The Interquartile Range VCAA Questions Continuous Distributions Section 8: The Normal Distribution Graphs of the Normal Distribution Calculating Probabilities Using Confidence Intervals Calculating Probabilities Using Technology The Standard Normal Distribution The Inverse Normal Distribution VCAA Questions Normal Distribution Integration and its Applications INTEGRATION AND ITS APPLICATIONS Integrating Algebraic Expressions Integrating Expressions General Approach n Integrating ( ax b) Integrating Trigonometric Expressions Integrating Exponential Expressions 1 Integrating x Integrating g'( x) gx ( ) Integrating Hybrid Functions Integrating Composite Functions Applications in Integration Solving for the Constant c Integration by Recognition Definite Integrals Important Properties of the Definite Integral Approximating the Area Under a Curve The Left Endpoint Approximation The Right Endpoint Approximation Using Integration to Find Areas Under Curves Changing the Sign of an Area Calculating Unsigned Areas or Areas Alternate Notations to Describe Areas Solving for an Unknown Given an Area or Definite Integral Areas Between Curves

4 INTEGRATION APPLICATIONS & STATISTICAL INFERENCE Applications in Integration (Continued) The Average Value of a Function Graphs of the Antiderivative Function Applications of Calculus to Motion in a Straight Line The Fundamental Theorem of Calculus Revisited Statistical Inference Population and Samples Parameters and Statistics Random Sampling Important Definitions What is Sampling? The Sampling Process Probability Sampling Cluster Sampling Simple Random Sampling Stratified Random Sampling Systematic Random Sampling Non-Probability Sampling Convenience Sampling Judgement Sampling Quota Sampling Accuracy, Bias, Variability and Precision Accuracy vs. Precision Accuracy vs Bias Bias vs Precision Accuracy, Bias vs Precision Bias vs Variability Errors in Sampling Sampling Errors Non-Sampling Errors Populations and Samples Common Notations Population and Sample Proportions Sampling Distribution of the Sample Proportion Sampling from Small Populations Sampling from Large Populations The Standard Error

5 Confidence Intervals for the Population Proportion Point Estimates Confidence Intervals Confidence Levels Critical Values Constructing Confidence Intervals Equivalent Rules Confidence Levels and Multipliers Cumulative Percentages and Critical Values Finding a Confidence Interval for P The Margin of Error Finding the Point Estimate and the Margin of Error from a Confidence Interval Factors that Affect the width of Confidence Levels Factors that Affect the Margin of Error Certainty vs Precision Calculating the Sample Size Needed to Estimate the Population Proportion