STAT 302 Introduction to Probability Learning Outcomes. Textbook: A First Course in Probability by Sheldon Ross, 8 th ed.

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1 STAT 302 Introduction to Probability Learning Outcomes Textbook: A First Course in Probability by Sheldon Ross, 8 th ed. Chapter 1: Combinatorial Analysis Demonstrate the ability to solve combinatorial problems o Use the basic principle of counting to obtain the total number of possible outcomes in a random experiment o Differentiate between permutations and combinations in a particular context o Explain if the order of outcomes matters in the context of the counting problem o Apply rules on permutations and combinations in solving counting problems Chapter 2: Axioms of Probability Demonstrate an understanding of basic probability concepts o Recognize the random experiment of interest in a given scenario o List all possible outcomes in the sample space of a random experiment o Recall the definitions of an event, the complement of an event, unions and intersections of events o Recall the meaning of a null set, and that of an event being contained in another event o Use Venn Diagrams to depict single events, complementary events, unions and intersections of a collection of events o Explain whether a collection of events are mutually exclusive o Apply the Commutative, Associative, Distributive and DeMorgan s laws to unions and intersections of events o Recall the probability of an event is a long-run relative frequency of occurrences of the event in repetitions of a random experiment o Recall the axioms of probability of an event o Define appropriate events in solving probability problems o Recognize if outcomes in a sample space are equally likely o Determine the probability of an event where the outcomes in a sample space are equally likely o Apply probability rules (complement rule, addition/general addition rule and others) in solving probability problems Chapter 3: Conditional Probability and Independence Apply the concepts of conditional probabilities and independence in computing probabilities of interest o Recall the definition and properties of conditional probabilities

2 o Compute conditional probabilities where probabilities of events in the conditional probability formula are given o Derive Bayes formula o Apply Bayes formula in solving probability problems o Recall the definition of the odds of an event o Interpret the value of the odds in terms of the relative probability of an event and its complement o Recall what is meant by independent events o Apply the definition of independence to determine whether two events or a collection of events are independent o Apply the definition of conditional independence to determine whether two events are conditionally independent Chapter 4: Discrete Random Variables Demonstrate an understanding of the basic concepts of discrete random variables and a number of common discrete s o Recall that a random variable is a function that maps outcomes in a sample space to a numerical quantity o Identify the random variable(s) of interest in a given scenario o Tell whether a random variable is discrete or not o Recall the definition and properties of the probability mass function of a discrete random variable o Recall the definition and properties of the cumulative function of a discrete random variable o Obtain the probability mass function and cumulative function for a discrete random variable of interest o Calculate probabilities associated with a discrete random variable o Calculate the expected value of a discrete random variable and that of a real function of a discrete variable o Interpret the expected value of a discrete random variable o Calculate the variance and standard deviation of a discrete random variable o Apply general properties of expectation and variance operators o Recall the definitions of a Bernoulli trial and a Binomial experiment o Recall the properties of a Poisson process o Recognize the Poisson random variable associated with a Poisson process o Recognize cases where the following s could be an applied model: Bernoulli, Binomial, Geometric, Negative Binomial, Poisson and Hypergeometric o Identify the parameters for the following s: Bernoulli, Binomial, Geometric, Negative Binomial, Poisson and Hypergeometric

3 o Calculate probabilities, the mean and variance of the following random variables: Bernoulli, Binomial, Geometric, Negative Binomial, Poisson and Hypergeometric o Approximate Binomial probabilities using a Poisson where appropriate Chapter 5: Continuous Random Variables Demonstrate an understanding of the basic concepts of continuous random variables and a number of common continuous s o Identify the random variable(s) of interest in a given scenario o Differentiate between discrete and continuous random variables o Recall the properties of the probability density function and cumulative function of a continuous random variable o Calculate probabilities of a continuous random variable from a given probability density function o Recognize that the probability that a continuous random variable whose value falls in a certain region is given by the area under the probability density function over that region o Recall the relationship between the probability density function and the cumulative function of a continuous random variable o Obtain the cumulative function from a probability density function for a continuous random variable, and vice versa o Calculate the expected value of a continuous random variable and that of a real function of a continuous variable o Calculate the variance and standard deviation of a continuous random variable o Apply general properties of expectation and variance operators o Recognize cases where the following s could be an applied model: Uniform, Normal, Exponential, Gamma and Beta o Identify the parameters for the following s: Uniform, Normal, Exponential, Gamma and Beta o Describe how the probability density function changes with the parameter(s) for the following s: Uniform, Normal, Exponential, Gamma and Beta o Calculate probabilities, the mean and variance of the following random variables: Uniform, Normal, Exponential, Gamma and Beta o Recall the properties of a Normal, and those of the standard Normal o Obtain probabilities related to Normal random variables using the standard Normal table o Approximate Binomial probabilities using a Normal where appropriate o Apply continuity correction when approximating Binomial probabilities using a Normal

4 o Recall that the time between two consecutive events that occur according to a Poisson process follows the Exponential o Explain the memoryless property of a continuous random variable, with the Exponential random variable as an example o Recall the relationship between the Exponential and the Gamma o Derive the cumulative function and the probability density function of a real function of a given continuous random variable Chapter 6: Jointly Distributed Random Variables Describe the joint and conditional s related to two or more discrete or continuous random variables o Recall the following definitions relating to two discrete random variables: joint probability mass function, marginal probability mass function and joint cumulative function o Recall the properties of the joint probability mass function of discrete random variables o Derive the joint probability mass function of two discrete random variables of interest in a given scenario o Obtain the marginal probability mass function and the joint cumulative function from the joint probability mass function of two discrete random variables o Recognize if two random variables are jointly continuous o Recall the properties of the joint probability density function of continuous random variables o Recall the definitions of the following of two continuous random variables: marginal probability density function and joint cumulative function o Recall the relationship between the joint probability density function and the joint cumulative function of two continuous random variables o Obtain the marginal probability density function and the joint cumulative function from the joint probability density function of two continuous random variables o Represent graphically the region on the x-y plane over which two jointly continuous random variables are defined o Set up appropriate bounds on the x-y plane and hence limits of integration for finding a probability of interest of two jointly continuous random variables o Identify the discrete or continuous random variables of interest in a given scenario o Compute probabilities related to two discrete or continuous random variables o Explain whether two random variables are independent o Derive the cumulative function and probability mass/density function of the sum of two independent random variables o Recall that the sum of two independent Gamma random variables with the same shape parameter also follows the Gamma

5 o Recall that Normality is preserved under linear combinations o Recall that the sum of independent Poisson random variables also follows the Poisson o Recall the definitions of the following for discrete random variables: conditional probability mass function and conditional cumulative function o Calculate conditional probability mass functions and conditional cumulative functions for bivariate discrete random variables o Determine if two discrete random variables are independent by using conditional probability mass function o Recall the definitions of the following for continuous random variables: conditional probability density function and conditional cumulative function o Calculate conditional probability density functions and conditional cumulative functions for bivariate continuous random variables o Determine if two continuous random variables are independent by using conditional probability density function o Calculate conditional probabilities of one variable given values of another variable by making use of conditional probability mass function or conditional probability density function Chapter 7: Properties of Expectation Apply properties of expectation to describe relationships between two random variables, and to compute probabilities, expectation and variance by conditioning. o Calculate the expectation of a real function of two random variables o Calculate the covariance and correlation between two random variables o Interpret the covariance and correlation between two random variables o Recall the properties of correlation between two random variables o Calculate the variance of a linear combination of random variables o Recall the properties of expectation for independent random variables o Calculate conditional expectations and conditional variances o Calculate the expectation of a random variable and that of a real function of a random variable using conditional expectation o Calculate the variance of a random variable using conditional expectation o Calculate the probability of an event by taking the conditional expectation of an indicator function of the event o Recognize situations where conditional expectation can be used to find expectations, variances and probabilities of interest o Recall the definition of the moment generating function of a random variable o Obtain the moment generating function of a random variable o Recall that the moment generating function provides a unique characterization of the of a random variable

6 o Identify the of a random variable and its parameters by comparing its moment generating function with those of common discrete and continuous s o Recall the definition of the nth moment of a random variable o Calculate the nth moment of a random variable by using moment generating function o Recall that for two independent random variables, the moment generating function of their sum is the product of their individual moment generating functions Chapter 8: Limit Theorems Apply Chebyshev s Inequality and the Central Limit Theorem in describing the of and calculating probabilities concerning general random variables, averages and sums of random variables o Recall the Markov s Inequality o Use the Markov s Inequality to prove the Chebyshev s Inequality o Apply the Chebyshev s Inequality to obtain a bound for the probability concerning a random variable given its mean and variance o Use the Chebyshev s Inequality to prove the Weak Law of Large Numbers o Recall the Weak Law of Large Numbers that states the convergence in probability of the average of random variables to their expected mean. o Apply the Central Limit Theorem to problems involving sums and averages of independent random variables from arbitrary s