**Instructional Objectives for ***Probability and Statistics for Engineers & Scientists* (Walpole)

*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.

**2.2 Events**

- 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*S*.^{2}

**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 S^{2}**

- Define the Gamma function and the Chi-squared distribution (from Section 6.7).
- Determine the distribution of the
*Χ*^{2}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
*S*^{2}. - Based on the resulting probabilities, judge the accuracy of an estimate of a population variance.

**8.6 t-Distribution**

- 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**

**9.3 Introduction**

- 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*σ*is known.^{2} - 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*σ*is not known.^{2}

**Instructional Objectives for ***Linear Algebra* (Levandosky)

*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 A
**x**=**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 R^{n}

^{n}

- State the properties of a subspace of a linear subspace of R
.^{n} - 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 R.^{n} - 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): R
^{n }--> R^{m}, 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(x
_{i})=Ax_{i}. - 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
^{-1}A C is a diagonal matrix.