Instructional Objectives for Probability and Statistics for Engineers & Scientists (Walpole)
Chapter 2 – Probability
2.1 Sample Space
- Define the terms: experiment, observation, sample space, and element (or sample point).
- Use a tree diagram to analyze an experiment.
- Use statements to describe experiments with very large number of elements.
- Define the terms: event, event complement, null set, intersection, mutually exclusive and union.
- Use Venn diagrams to analyze the relationships between events in a sample space.
2.3 Counting Sample Points
- Count sample points using the “multiplication rule” or “generalized multiplication rule”.
- State the difference between a permutation and a combination, and use factorials to solve counting problems using permutations and combinations.
- Determine the number of arrangements of objects in circular permutations.
- Define cells and use them to determine if a partition is achieved if a set is divided into cells.
2.4 Probability of an Event
- Define probability as it relates to events and sample spaces.
- Determine the probability of events in a sample space.
2.5 Additive Rules
- Use the additive rule and its corollaries to solve probability problems.
- State the additive rule for complementary events.
2.6 Conditional Probability, Independence and the Product Rule
- Use the definition of conditional probability to solve probability problems.
- Define independent events, and state how independence applies to conditional probability.
- State and use the product rule to solve probability problems.
2.7 Bayes’ Rule
- State the Theorem of Total Probability for the partition of a sample space.
- State and use Bayes’ rule to solve probability problems.
Chapter 3 – Random Variables and Probability Distributions
3.1 Concept of a Random Variable
- Define the terms random variable and Bernoulli random variable and give examples.
- Explain the difference between discrete and continuous sample spaces, and between discrete and continuous random variables.
3.2 Discrete Probability Distributions
- Determine the probability of values for a discrete random variable.
- State the conditions for a valid probability function, probability mass function or probability distribution for a discrete random variable.
- Define and determine the cumulative distribution function for a discrete random variable.
- Graph a probability mass function or cumulative distribution function for a discrete random variable.
3.3 Continuous Probability Distributions
- Determine the probability of values for a continuous random variable.
- State the conditions for a valid probability density function for a continuous random variable.
- Define and determine the cumulative distribution function for a continuous random variable.
- Graph a probability mass function or cumulative distribution function for a continuous random variable.
Chapter 4 – Mathematical Expectation
4.1 Mean of a Random Variable
- Calculate the expected value of a random variable, for both continuous and discrete cases.
- Calculate the expected value of a function of a random variable, for both continuous and discrete cases.
4.2 Variance and Covariance of Random Variables
- Calculate the variance and standard deviation of a random variable, for both continuous and discrete cases.
- Calculate the variance and standard deviation of a function of a random variable, for both continuous and discrete cases.
4.3 Means and Variances of Linear Combinations of Random Variables
- Demonstrate a familiarity with joint probability distribution functions and marginal distributions (from Section 3.4).
- Define the term covariance.
- Determine the mean and variance of a linear combination of random variables.
Chapter 5 – Some Discrete Probability Distributions
5.2 Binomial and Multinomial Distributions
- Define the terms Bernoulli process and Bernoulli trial and give examples.
- Determine the probability of values of random variables using the binomial distribution formula.
- Use the Binomial Probability Sums table (Table A.1) to solve binomial probability problems.
- Calculate the mean and variance of a binomial distribution.
5.5 Poisson Distribution and the Poisson Process
- State the properties of a Poisson Process.
- Determine the probability of values of random variables using the Poisson distribution formula.
- Use the Poisson Probability Sums table (Table A.2) to solve Poisson probability problems.
- Calculate the mean and variance of a Poisson distribution.
- Solve problems in which the Binomial distribution is approximated by the Poisson distribution.
Chapter 6 – Some Continuous Probability Distributions
6.1 Continuous Uniform Distribution
- Determine the pdf for a continuous uniform random variable.
- Calculate the mean and variance of a uniform distribution.
- Use a sketch of the shape of the uniform distribution to set up related problems.
6.2 Normal Distribution
- Determine the pdf for a normal (Gaussian) random variable.
- Demonstrate familiarity with the properties of the normal curve.
- Describe how the shape of the normal curve changes if mean value and/or variance changes.
- Calculate the mean and variance of a normal (Gaussian) distribution.
6.3 Areas under the Normal Curve
- For problem involving a normally distributed random variable X, transform the normal random variable to a standard normal random variable Z.
- Use a sketch of the shape of a normal distribution to set up related problems.
- Using the properties of the standard normal curve and the Normal Probability Table (Table A.3), calculate probabilities associated with a Gaussian random variable.
- Given the probability of a normal random variable (X), find the value(s) of X that result in that probability.
6.4 Applications of the Normal Distribution
- Solve normal distribution problems that involve a finite precision in the measurement of the random variable.
- Use a sketch of the shape of the normal distribution to set up related problems.
6.5 Normal Approximation to the Binomial
- Describe the conditions under which a binomial distribution may be approximated with a normal distribution.
- Solve problems involving approximating the binomial distribution with a normal distribution.
6.6 Gamma and Exponential Distributions
- Determine the pdf for a random variable that has an exponential distribution.
- Use a sketch of the shape of the exponential distribution to set up related problems.
- Solve problems where the probability of a random variable is exponential, and this probability is used as the probability of success in Bernoulli trials.
Chapter 8 – Fundamental Sampling Distributions and Data Descriptions
8.1 Random Sampling
- Define the terms population and sample of a population and give examples.
- Define bias, and state how bias is eliminated when sampling a population.
8.2 Some Important Statistics
- Define the terms statistic, sample mean, sample median, sample mode, and sample variance.
- Given a sample of data, compute the statistics of the sample.
8.3 Sampling Distributions
- Define the term sampling distribution as it relates to and S2.
8.4 Sampling Distribution of Means and the Central Limit Theorem
- For the sampling distribution of , determine the mean and variance.
- State the Central Limit Theorem, and define the parameters of the distribution of the statistic.
- State the minimum sample size for which the Central Limit Theorem applies.
- Solve problems involving the sampling distribution of the means.
- Solve problems involving the sampling distribution of the difference between two means.
8.5 Sampling Distribution of S2
- Define the Gamma function and the Chi-squared distribution (from Section 6.7).
- Determine the distribution of the Χ2 statistic as it relates to the variance of a sample of size n.
- Use a sketch of the shape of the chi-squared distribution to set up related problems.
- Use the Chi-Squared Distribution Probability Table (Table A.5) to solve problems involving the sampling distribution of S2.
- Based on the resulting probabilities, judge the accuracy of an estimate of a population variance.
- Describe the circumstances under which the t-distribution is used, and state the formula for the T statistic.
- Use a sketch of the shape of the t-distribution to set up related problems.
- Use the Student t-Distribution Probability Table (Table A.4) to solve problems involving a population mean.
- Based on the resulting probabilities, judge the accuracy of an estimate of a population mean.
Chapter 9 – One- and Two-Sample Estimation Problems
- Define the terms population, parameter, sample, statistic and point estimate as they relate to classical methods of estimation.
- State the conditions under which sample statistic is an unbiased estimator of population parameter θ.
- Given a number of unbiased estimators of a population parameter, determine the most efficient estimator, and state what makes one unbiased estimator more efficient than another.
- Define the terms interval estimate and confidence interval.
9.4 Single Sample: Estimating the Mean
- Determine the upper and lower limits of a confidence interval in estimating μ with a single sample if σ2 is known.
- Determine the sample size needed for a given maximum error in a point estimate of a population mean.
- Determine one-sided confidence bounds of a confidence interval.
- Determine the upper and lower limits of a confidence interval in estimating μ with a single sample if σ2 is not known.
Instructional Objectives for Linear Algebra (Levandosky)
Chapter 3 – Linear Independence
- Define linear independence.
- Apply Gaussian elimination to a set of vectors to determine if they are linearly independent or not.
Chapter 4 – Dot Products and Cross Products
- Determine the dot product of two vectors with the same number of elements; use the dot product to determine if the two vectors are perpendicular.
- Calculate the length (a.k.a. magnitude or norm) of a given vector, and determine a unit vector in the same direction as the given vector.
- Apply the properties of vector magnitudes given in Proposition 4.2 (page 22) and 4.3 (Cauchy-Schwarz Inequality, page 23).
Chapter 5 – Systems of Linear Equations
- Using Gaussian elimination, determine if a system of linear equations has a unique solution, an infinite number of solutions, or no solution.
- Define the terms consistent and inconsistent as they apply to systems of linear equations.
Chapter 6 – Matrices
- For a system of linear equations, identify its coefficient matrix and its augmented matrix.
- Describe the three row operations that can be performed on an augmented matrix without changing the solutions to the original linear system of equations.
- State the format requirements for an augmented matrix in reduced row echelon form (rref); define the terms pivot, pivot variables and free variables.
- Perform row reduction operations on an augmented matrix to reduce it to rref, then determine the solutions to the corresponding system of linear equations.
Chapter 7 – Matrix-Vector Products
- Define the transpose of a vector.
- Calculate the matrix-vector product of matrix A and vector v.
- State the properties of matrix-vector products in Proposition 7.1 (p. 49).
Chapter 8 – Null Space
- Define homogeneous and inhomogeneous as they apply to a system of linear equations.
- For a set of vectors all with the same dimensions, define the span of the set of vectors.
- Define the null space of matrix A, written as N(A), and for a given matrix A, determine it's null space.
Chapter 9 – Column Space
- Define the column space of matrix A, written as C(A), and for a given matrix A, determine it's column space.
- For the system of linear equations described by Ax=b, determine the conditions under which b is in the column space of A (b is in C(A)), such that Ax+b has a solution.
Chapter 10 – Subspaces of Rn
- State the properties of a subspace of a linear subspace of Rn.
- Determine if a set of vectors or solutions of a system of linear equations is a subspace.
Chapter 11 – Basis for a Subspace
- Define the standard basis for Rn.
- Given a matrix A, determine a basis for its null space.
- Given a matrix A, determine a basis for its column space.
Chapter 12 – Dimension of a Subspace
- Define what is meant by the dimension of a subspace.
- Given an m x n matrix A, state the relation between rank(A), nullity(A) and n.
- Given a matrix A, determine rref(A) and use that to determine its rank and nullity.
Chapter 13 – Linear Transformations
- State the conditions under which a function is considered a linear transform.
- Given a function T(x): Rn --> Rm, determine if it is a linear transform.
- Given input and output vectors of a linear transform T(x), determine its transform matrix A such that T(xi)=Axi.
- Use the properties specified in Proposition 13.5 (page 86) to solve linear transformation problems.
Chapter 14 – Examples of Linear Transformations
- Determine the linear transformation matrix for an identity or scaling transform.
- Determine the linear transformation matrix for a 2D or 3D rotation of a vector around the x-axis by an angle of θ.
- Define the term diagonal matrix.
- Given a vector that spans a line in 2D or 3D, determine the transformation matrix A that can be used to compute the projection of another vector onto that line.
Chapter 15 – Composition and Matrix Multiplication
- Given equations for f(x,y) and g(x,y), determine the equation for f ° g = f(g(x,y)).
- State the conditions under which matrix multiplication of two matrices A and B is possible, and determine the dimensions of their product.
- Determine the transformation matrix for f ° g, where f and g are linear transformations.
Chapter 16 – Inverses
- Define what is meant by an invertible function, and state state the definitions of onto and one-to-one.
- For a linear transform, state the conditions for invertibility related to the transform matrix A, including the dimensions of A, rank of A, and rref(A).
- Determine the inverse of an n x n matrix A by augmenting it with the nxn identity matrix, then reducing that to reduced row echelon form.
- Determine the inverse of a 2 x 2 matrix using the method introduced on page 110.
Chapter 17 – Determinants
- Calculate the determinant of a 2 x 2 matrix using the formula provided in the chapter.
- Calculate the determinant of an n x n matrix by expanding along a row or column.
- Define upper triangular and lower triangular as they apply to square matrices.
- Calculate the determinant of an n x n matrix using Gaussian elimination.
Chapter 18 – Transpose of a Matrix
- Determine the transpose of an m x n matrix.
- Solve problems involving the transpose of a matrix using the properties of transposes given in Proposition 18.1.
Chapter 23 – Eigenvectors
- Given an n x n matrix, determine its characteristic polynomial.
- Given an n x n matrix, determines its eigenvalues and a corresponding eigenvector for each.
- State the conditions under which two matrices are similar.
- Given a diagonalizable n x n matrix A, determine matrix C and matrix D so that D = C-1A C is a diagonal matrix.