Electrical and Computer Engineering Department

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

### Chapter 2–Probability

2.1 Sample Space

1. Define the terms: experiment, observation, sample space, and element (or sample point).
2. Use a tree diagram to analyze an experiment.
3. Use statements to describe experiments with very large number of elements.

2.2 Events

1. Define the terms: event, event complement, null set, intersection, mutually exclusive and union.
2. Use Venn diagrams to analyze the relationships between events in a sample space.

2.3 Counting Sample Points

1. Count sample points using the “multiplication rule” or “generalized multiplication rule”.
2. State the difference between a permutation and a combination, and use factorials to solve counting problems using permutations and combinations.
3. Determine the number of arrangements of objects in circular permutations.
4. 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

1. Define probability as it relates to events and sample spaces.
2. Determine the probability of events in a sample space.

1. Use the additive rule and its corollaries to solve probability problems.
2. State the additive rule for complementary events.

2.6 Conditional Probability, Independence and the Product Rule

1. Use the definition of conditional probability to solve probability problems.
2. Define independent events, and state how independence applies to conditional probability.
3. State and use the product rule to solve probability problems.

2.7 Bayes’ Rule

1. State the Theorem of Total Probability for the partition of a sample space.
2. State and use Bayes’ rule to solve probability problems.

### Chapter 3 – Random Variables and Probability Distributions

3.1 Concept of a Random Variable

1. Define the terms random variable and Bernoulli random variable and give examples.
2. Explain the difference between discrete and continuous sample spaces, and between discrete and continuous random variables.

3.2 Discrete Probability Distributions

1. Determine the probability of values for a discrete random variable.
2. State the conditions for a valid probability function, probability mass function or probability distribution for a discrete random variable.
3. Define and determine the cumulative distribution function for a discrete random variable.
4. Graph a probability mass function or cumulative distribution function for a discrete random variable.

3.3 Continuous Probability Distributions

1. Determine the probability of values for a continuous random variable.
2. State the conditions for a valid probability density function for a continuous random variable.
3. Define and determine the cumulative distribution function for a continuous random variable.
4. 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

1. Calculate the expected value of a random variable, for both continuous and discrete cases.
2. 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

1. Calculate the variance and standard deviation of a random variable, for both continuous and discrete cases.
2. 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

1. Demonstrate a familiarity with joint probability distribution functions and marginal distributions (from Section 3.4).
2. Define the term covariance.
3. Determine the mean and variance of a linear combination of random variables.

### Chapter 5 – Some Discrete Probability Distributions

5.2 Binomial and Multinomial Distributions

1. Define the terms Bernoulli process and Bernoulli trial and give examples.
2. Determine the probability of values of random variables using the binomial distribution formula.
3. Use the Binomial Probability Sums table (Table A.1) to solve binomial probability problems.
4. Calculate the mean and variance of a binomial distribution.

5.5 Poisson Distribution and the Poisson Process

1. State the properties of a Poisson Process.
2. Determine the probability of values of random variables using the Poisson distribution formula.
3. Use the Poisson Probability Sums table (Table A.2) to solve Poisson probability problems.
4. Calculate the mean and variance of a Poisson distribution.
5. Solve problems in which the Binomial distribution is approximated by the Poisson distribution.

### Chapter 6 – Some Continuous Probability Distributions

6.1 Continuous Uniform Distribution

1. Determine the pdf for a continuous uniform random variable.
2. Calculate the mean and variance of a uniform distribution.
3. Use a sketch of the shape of the uniform distribution to set up related problems.

6.2 Normal Distribution

1. Determine the pdf for a normal (Gaussian) random variable.
2. Demonstrate familiarity with the properties of the normal curve.
3. Describe how the shape of the normal curve changes if mean value and/or variance changes.
4. Calculate the mean and variance of a normal (Gaussian) distribution.

6.3 Areas under the Normal Curve

1. For problem involving a normally distributed random variable X, transform the normal random variable to a standard normal random variable Z.
2. Use a sketch of the shape of a normal distribution to set up related problems.
3. Using the properties of the standard normal curve and the Normal Probability Table (Table A.3), calculate probabilities associated with a Gaussian random variable.
4. 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

1. Solve normal distribution problems that involve a finite precision in the measurement of the random variable.
2. Use a sketch of the shape of the normal distribution to set up related problems.

6.5 Normal Approximation to the Binomial

1. Describe the conditions under which a binomial distribution may be approximated with a normal distribution.
2. Solve problems involving approximating the binomial distribution with a normal distribution.

6.6 Gamma and Exponential Distributions

1. Determine the pdf for a random variable that has an exponential distribution.
2. Use a sketch of the shape of the exponential distribution to set up related problems.
3. 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

1. Define the terms population and sample of a population and give examples.
2. Define bias, and state how bias is eliminated when sampling a population.

8.2 Some Important Statistics

1. Define the terms statistic, sample mean, sample median, sample mode, and sample variance.
2. Given a sample of data, compute the statistics of the sample.

8.3 Sampling Distributions

1. Define the term sampling distribution as it relates to  and S2.

8.4 Sampling Distribution of Means and the Central Limit Theorem

1. For the sampling distribution of , determine the mean and variance.
2. State the Central Limit Theorem, and define the parameters of the distribution of the statistic.
3. State the minimum sample size for which the Central Limit Theorem applies.
4. Solve problems involving the sampling distribution of the means.
5. Solve problems involving the sampling distribution of the difference between two means.

8.5 Sampling Distribution of S2

1. Define the Gamma function and the Chi-squared distribution (from Section 6.7).
2. Determine the distribution of the Χ2 statistic as it relates to the variance of a sample of size n.
3. Use a sketch of the shape of the chi-squared distribution to set up related problems.
4. Use the Chi-Squared Distribution Probability Table (Table A.5) to solve problems involving the sampling distribution of S2.
5. Based on the resulting probabilities, judge the accuracy of an estimate of a population variance.

8.6 t-Distribution

1. Describe the circumstances under which the t-distribution is used, and state the formula for the T statistic.
2. Use a sketch of the shape of the t-distribution to set up related problems.
3. Use the Student t-Distribution Probability Table (Table A.4) to solve problems involving a population mean.
4. 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

1. Define the terms population, parameter, sample, statistic and point estimate as they relate to classical methods of estimation.
2. State the conditions under which sample statistic  is an unbiased estimator of population parameter θ.
3. 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.
4. Define the terms interval estimate and confidence interval.

9.4 Single Sample: Estimating the Mean

1. Determine the upper and lower limits of a confidence interval in estimating μ with a single sample if σ2 is known.
2. Determine the sample size needed for a given maximum error in a point estimate of a population mean.
3. Determine one-sided confidence bounds of a confidence interval.
4. 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

1. Define linear independence.
2. Apply Gaussian elimination to a set of vectors to determine if they are linearly independent or not.

### Chapter 4 – Dot Products and Cross Products

1. 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.
2. 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.
3. 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

1. Using Gaussian elimination, determine if a system of linear equations has a unique solution, an infinite number of solutions, or no solution.
2. Define the terms consistent and inconsistent as they apply to systems of linear equations.

### Chapter 6 – Matrices

1. For a system of linear equations, identify its coefficient matrix and its augmented matrix.
2. Describe the three row operations that can be performed on an augmented matrix without changing the solutions to the original linear system of equations.
3. State the format requirements for an augmented matrix in reduced row echelon form (rref); define the terms pivot, pivot variables and free variables.
4. 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

1. Define the transpose of a vector.
2. Calculate the matrix-vector product of matrix A and vector v.
3. State the properties of matrix-vector products in Proposition 7.1 (p. 49).

### Chapter 8 – Null Space

1. Define homogeneous and inhomogeneous as they apply to a system of linear equations.
2. For a set of vectors all with the same dimensions, define the span of the set of vectors.
3. 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

1. Define the column space of matrix A, written as C(A), and for a given matrix A, determine it's column space.
2. 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

1. State the properties of a subspace of a linear subspace of Rn.
2. Determine if a set of vectors or solutions of a system of linear equations is a subspace.

### Chapter 11 – Basis for a Subspace

1. Define the standard basis for Rn.
2. Given a matrix A, determine a basis for its null space.
3. Given a matrix A, determine a basis for its column space.

### Chapter 12 – Dimension of a Subspace

1. Define what is meant by the dimension of a subspace.
2. Given an m x n matrix A, state the relation between rank(A), nullity(A) and n.
3. Given a matrix A, determine rref(A) and use that to determine its rank and nullity.

### Chapter 13 – Linear Transformations

1. State the conditions under which a function is considered a linear transform.
2. Given a function T(x): R--> Rm, determine if it is a linear transform.
3. Given input and output vectors of a linear transform T(x), determine its transform matrix A such that T(xi)=Axi.
4. Use the properties specified in Proposition 13.5 (page 86) to solve linear transformation problems.

### Chapter 14 – Examples of Linear Transformations

1. Determine the linear transformation matrix for an identity or scaling transform.
2. Determine the linear transformation matrix for a 2D or 3D rotation of a vector around the x-axis by an angle of θ.
3. Define the term diagonal matrix.
4. 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

1. Given equations for f(x,y) and g(x,y), determine the equation for f ° g = f(g(x,y)).
2. State the conditions under which matrix multiplication of two matrices A and B is possible, and determine the dimensions of their product.
3. Determine the transformation matrix for f ° g, where f and g are linear transformations.

### Chapter 16 – Inverses

1. Define what is meant by an invertible function, and state state the definitions of onto and one-to-one.
2. 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).
3. 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.
4. Determine the inverse of a 2 x 2 matrix using the method introduced on page 110.

### Chapter 17 – Determinants

1. Calculate the determinant of a 2 x 2 matrix using the formula provided in the chapter.
2. Calculate the determinant of an n x n matrix by expanding along a row or column.
3. Define upper triangular and lower triangular as they apply to square matrices.
4. Calculate the determinant of an n x n matrix using Gaussian elimination.

### Chapter 18 – Transpose of a Matrix

1. Determine the transpose of an m x n matrix.
2. Solve problems involving the transpose of a matrix using the properties of transposes given in Proposition 18.1.

### Chapter 23 – Eigenvectors

1. Given an n x n matrix, determine its characteristic polynomial.
2. Given an n x n matrix, determines its eigenvalues and a corresponding eigenvector for each.
3. State the conditions under which two matrices are similar.
4. Given a diagonalizable n x n matrix A, determine matrix C and matrix D so that D = C-1A C is a diagonal matrix.