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.

2.5 Additive Rules

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 (Schaum’s Outline) (Lipschutz & Lipson)

## Lesson #32: Vectors, Matrices, Matrix Algebra

1. Demonstrate knowledge of how row and column vectors are structured, and perform vector addition, subtraction, and scalar multiplication.
2. Define linear combination as it applies to vectors.
3. For vectors, compute dot (inner) products, norms (lengths) and projections of one vector onto another.
4. Determine if two vectors are orthogonal. Calculate unit vectors, and determine if two vectors are orthonormal.
5. Perform matrix scalar multiplication, matrix addition and matrix multiplication.

## Lesson #33: Square Matrices, Special Matrices

1. Identify the main diagonal of a square matrix, and compute the trace.
2. Define what is meant by the identity matrix, and perform matrix multiplication with it.
3. Perform computations of square matrices input to polynomials.
4. Calculate the inverse of a 2x2 matrix, if it exists. State the conditions for the inverse to exist.
5. Define diagonal matrix, symmetric matrix, orthogonal matrix, and normal

## Lesson #34: Systems of Linear Equations

1. Define the terms linear equation and systems of linear equations.
2. State the conditions under which a system of linear equations is homogeneous.
3. Determine if an input vector is a solution to a set of linear equations.
4. Define the terms inconsistent and consistent as they relate to systems of linear equations, and state the two cases of a consistent system of linear equations.
5. Considering a system of linear equations in triangular form, define what is meant by echelon form, pivot variables, free variables and dimension.
6. Solve systems of linear equations using back substitution (when the system is given in triangular form), or Gaussian elimination to put the system in triangular form, then solving using back substitution.
7. Form the augmented matrix from the coefficients and constants of a system of linear equations.
8. Determine the row echelon form of a matrix, and the row canonical form (also called reduced row-echelon form).
9. Create an augmented matrix from a system of linear equations, and use Gaussian elimination on the matrix to solve the system.

## Lesson #35: Homogeneous Systems, Elementary Matrices

1. Write one vector as a linear combination of other vectors by transforming the problem into a system of linear equations, then solving the system.
2. Determine a basis for the general solution of a homogeneous system.
3. Calculate the inverse of a matrix, if it exists, using Elementary matrices.

## Lesson #36: Linear Independence, Rank of a Matrix

1. Determine if a set of vectors is linearly independent or not.
2. Define rank of a matrix, and calculate it for a given matrix.

## Lesson #38: Determinants, Properties of Determinants

1. Calculate the determinant of a square matrix using the method presented in Figure 8-1, or the alternate method.
2. Calculate the determinant of a square matrix that has a row or column of all zeros.
3. Given square matrix A, calculate its determinant by using row reduction techniques to transform it into triangular matrix B, while accounting for how the row reduction techniques affect how |B| relates to |A|.

## Lesson #39: Minors, Cofactors and Cramer’s Rule

1. Define the terms minor, signed minor and cofactor.
2. Evaluate the determinant of a square matrix using Laplace expansion.
3. Use Cramer’s rule to solve a system of linear equations.

## Lesson #40: Eigenvalues/Eigenvectors

1. Calculate the characteristic polynomial for a square matrix A using |lIn – A|.
2. Calculate the characteristic polynomial for a 2x2 or 3x3 matrix A using the specific formulas provided that use |A| and trace(A).
3. Using the characteristic polynomial, determine the eigenvalues for a square matrix.
4. State the mathematical relationship between a square matrix, its eigenvalues and its eigenvectors.

## Lesson #41: Eigenvalues/Eigenvectors

1. Compute the eigenvalues and eigenvectors for an n x n square matrix using the Diagonalization Algorithm (Algorithm 9.1). Describe the effect if there are fewer than n non-zero eigenvalues.
2. For diagonalizable square matrix A, find the matrix P and the diagonal matrix D such that D=P-1
3. Compute a polynomial function of square matrix A using diagonalization to simplify calculations.

## Lesson #43: Diagonalization

1. Compute the eigenvalues and eigenvectors for an n x n square symmetric matrix using the Orthogonal Diagonalization Algorithm (Algorithm 9.2).

## Lesson #44: ECE Applications of Linear Algebra & MATLAB

1. Solve a system of linear equations by using the MATLAB rref function to place the augmented matrix in reduced row echelon form.
2. Given a square matrix, use MATLAB to determine its eigenvalues and a set of orthonormal eigenvectors.