Boundless Lecture Slides Available on the Boundless Teaching

Boundless Lecture Slides Available on the Boundless Teaching Platform Free to share, print, make copies and changes. Get yours at www. boundless. com

Boundless Teaching Platform Boundless empowers educators to engage their students with affordable, customizable textbooks and intuitive teaching tools. The free Boundless Teaching Platform gives educators the ability to customize textbooks in more than 20 subjects that align to hundreds of popular titles. Get started by using high quality Boundless books, or make switching to our platform easier by building from Boundless content pre-organized to match the assigned textbook. This platform gives educators the tools they need to assign readings and assessments, monitor student activity, and lead their classes with pre-made teaching resources. Using Boundless Presentations The Appendix The appendix is for you to use to add depth and breadth to your lectures. You can simply drag and drop slides from the appendix into the main presentation to make for a richer lecture experience. Get started now at: http: //boundless. com/teaching-platform Free to edit, share, and copy Feel free to edit, share, and make as many copies of the Boundless presentations as you like. We encourage you to take these presentations and make them your own. If you have any questions or problems please email: educators@boundless. com Free to share, print, make copies and changes. Get yours at www. boundless. com

About Boundless is an innovative technology company making education more affordable and accessible for students everywhere. The company creates the world’s best open educational content in 20+ subjects that align to more than 1, 000 popular college textbooks. Boundless integrates learning technology into all its premium books to help students study more efficiently at a fraction of the cost of traditional textbooks. The company also empowers educators to engage their students more effectively through customizable books and intuitive teaching tools as part of the Boundless Teaching Platform. More than 2 million learners access Boundless free and premium content each month across the company’s wide distribution platforms, including its website, i. OS apps, Kindle books, and i. Books. To get started learning or teaching with Boundless, visit boundless. com. Free to share, print, make copies and changes. Get yours at www. boundless. com

Probability and Variability Discrete Random Variables The Binomial Random Variable Other Random Variables Probability and Variability Free to share, print, make copies and changes. Get yours at www. boundless. com

Probability and Variability > Discrete Random Variables • Two Types of Random Variables • Probability Distributions for Discrete Random Variables • Expected Values of Discrete Random Variables Free to share, print, make copies and changes. Get yours at www. boundless. com/statistics/textbooks/boundless-statistics-textbook/probability-and-variability-9/discrete-random-variables-36/

Probability and Variability > The Binomial Random Variable • The Binomial Formula • Binomial Probability Distributions • Mean, Variance, and Standard Deviation of the Binomial Distribution • Additional Properties of the Binomial Distribution Free to share, print, make copies and changes. Get yours at www. boundless. com/statistics/textbooks/boundless-statistics-textbook/probability-and-variability-9/the-binomial-random-variable-37/

Probability and Variability > Other Random Variables • The Poisson Random Variable • The Hypergeometric Random Variable Free to share, print, make copies and changes. Get yours at www. boundless. com/statistics/textbooks/boundless-statistics-textbook/probability-and-variability-9/other-random-variables-38/

Appendix Free to share, print, make copies and changes. Get yours at www. boundless. com

Probability and Variability Key terms • Bernoulli Trial an experiment whose outcome is random and can be either of two possible outcomes, "success" or "failure" • binomial distribution the discrete probability distribution of the number of successes in a sequence of independent yes/no experiments, each of which yields success with probability • central limit theorem a theorem which states that, given certain conditions, the mean of a sufficiently large number of independent random variables--each with a well-defined mean and well-defined variance-- will be approximately normally distributed • continuous random variable obtained from data that can take infinitely many values • covariance A measure of how much two random variables change together. • discrete random variable obtained by counting values for which there are no in-between values, such as the integers 0, 1, 2, …. • disjoint having no members in common; having an intersection equal to the empty set. • expected value of a discrete random variable, the sum of the probability of each possible outcome of the experiment multiplied by the value itself • factorial The result of multiplying a given number of consecutive integers from 1 to the given number. In equations, it is symbolized by an exclamation mark (!). For example, 5! = 1 * 2 * 3 * 4 * 5 = 120. Free to share, print, make copies and changes. Get yours at www. boundless. com

Probability and Variability • floor function maps a real number to the smallest following integer • hypergeometric distribution a discrete probability distribution that describes the number of successes in a sequence of draws from a finite population without replacement • mean one measure of the central tendency either of a probability distribution or of the random variable characterized by that distribution • median the numerical value separating the higher half of a data sample, a population, or a probability distribution, from the lower half • Mode the value that appears most often in a set of data • Poisson distribution A discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time and/or space, if these events occur with a known average rate and independently of the time since the last event. • probability distribution A function of a discrete random variable yielding the probability that the variable will have a given value. • probability mass function a function that gives the relative probability that a discrete random variable is exactly equal to some value • probability mass function a function that gives the probability that a discrete random variable is exactly equal to some value • random variable a quantity whose value is random and to which a probability distribution is assigned, such as the possible outcome of a roll of a die • standard deviation shows how much variation or dispersion exists from the average (mean), or expected value • variance a measure of how far a set of numbers is spread out Free to share, print, make copies and changes. Get yours at www. boundless. com

Probability and Variability Probability Mass Function An important formula in understanding binomial distribution. Free to share, print, make copies and changes. Get yours at www. boundless. com Wikipedia. "Binomial random variable. " GNU FDL http: //en. wikipedia. org/wiki/Binomial_random_variable View on Boundless. com

Probability and Variability Probability Density Function The image shows the probability density function (pdf) of the normal distribution, also called Gaussian or "bell curve", the most important continuous random distribution. As notated on the figure, the probabilities of intervals of values corresponds to the area under the curve. Free to share, print, make copies and changes. Get yours at www. boundless. com Wikipedia. "Standard deviation diagram. " CC BY-SA http: //en. wikipedia. org/wiki/File: Standard_deviation_diagram. svg View on Boundless. com

Probability and Variability Probability Histogram This histogram displays the probabilities of each of the three discrete random variables. Free to share, print, make copies and changes. Get yours at www. boundless. com Wikidot. "(Discrete) Probability Distributions - Statistics. " CC BY-SA http: //statistics. wikidot. com/ch 6 View on Boundless. com

Probability and Variability Probability Mass Function This shows the graph of a probability mass function. All the values of this function must be non-negative and sum up to 1. Free to share, print, make copies and changes. Get yours at www. boundless. com Wikipedia. "Discrete probability distrib. " CC BY-SA http: //en. wikipedia. org/wiki/File: Discrete_probability_distrib. svg View on Boundless. com

Probability and Variability Poisson Probability Mass Function The horizontal axis is the index k, the number of occurrences. The function is only defined at integer values of k. The connecting lines are only guides for the eye. Free to share, print, make copies and changes. Get yours at www. boundless. com Wikimedia. "Poisson pmf. " CC BY http: //commons. wikimedia. org/wiki/File: Poisson_pmf. svg View on Boundless. com

Probability and Variability Discrete Probability Disrtibution This shows the probability mass function of a discrete probability distribution. The probabilities of the singletons {1}, {3}, and {7} are respectively 0. 2, 0. 5, 0. 3. A set not containing any of these points has probability zero. Free to share, print, make copies and changes. Get yours at www. boundless. com Wikipedia. "Discrete probability distrib. " CC BY-SA http: //en. wikipedia. org/wiki/File: Discrete_probability_distrib. svg View on Boundless. com

Probability and Variability Discrete Probability Distribution This table shows the values of the discrete random variable can take on and their corresponding probabilities. Free to share, print, make copies and changes. Get yours at www. boundless. com Wikidot. "(Discrete) Probability Distributions - Statistics. " CC BY-SA http: //statistics. wikidot. com/ch 6 View on Boundless. com

Probability and Variability Mode This formula is for calculating the mode of a binomial distribution. Free to share, print, make copies and changes. Get yours at www. boundless. com Wikipedia. "Binomial distribution. " GNU FDL http: //en. wikipedia. org/wiki/Binomial_distribution View on Boundless. com

Probability and Variability Binomial Coefficient Formula for the binomial coefficient. Free to share, print, make copies and changes. Get yours at www. boundless. com Wikipedia. "Binomial random variable. " GNU FDL http: //en. wikipedia. org/wiki/Binomial_random_variable View on Boundless. com

Probability and Variability Covariance 3 The final formula for the covariance of a binomial distribution. Free to share, print, make copies and changes. Get yours at www. boundless. com Wikipedia. "Binomial distribution. " GNU FDL http: //en. wikipedia. org/wiki/Binomial_distribution View on Boundless. com

Probability and Variability Covariance 2 The next step in determining covariance. Free to share, print, make copies and changes. Get yours at www. boundless. com Wikipedia. "Binomial distribution. " GNU FDL http: //en. wikipedia. org/wiki/Binomial_distribution View on Boundless. com

Probability and Variability Floor Function Floor function is the lowest previous integer in a series. Free to share, print, make copies and changes. Get yours at www. boundless. com Wikipedia. "Binomial distribution. " GNU FDL http: //en. wikipedia. org/wiki/Binomial_distribution View on Boundless. com

Probability and Variability Covariance 1 The first part of finding covariance. Free to share, print, make copies and changes. Get yours at www. boundless. com Wikipedia. "Binomial distribution. " GNU FDL http: //en. wikipedia. org/wiki/Binomial_distribution View on Boundless. com

Probability and Variability Wind pollination These male (a) and female (b) catkins from the goat willow tree (Salix caprea) have structures that are light and feathery to better disperse and catch the wind-blown pollen. Free to share, print, make copies and changes. Get yours at www. boundless. com Open. Stax CNX. "Open. Stax College, Pollination and Fertilization. October 17, 2013. " CC BY 3. 0 http: //cnx. org/content/m 44723/latest/Figure_32_02_05. jpg View on Boundless. com

Probability and Variability Table 2 These are the probabilities of the 2 coin flips. Free to share, print, make copies and changes. Get yours at www. boundless. com Open. Stax CNX. "David Lane, Binomial Distribution. May 23, 2013. " CC BY 3. 0 http: //cnx. org/content/m 11024/latest/ View on Boundless. com

Probability and Variability Binomial Probability Distribution This is a graphic representation of a binomial probability distribution. Free to share, print, make copies and changes. Get yours at www. boundless. com Open. Stax CNX. "David Lane, Binomial Distribution. May 23, 2013. " CC BY 3. 0 http: //cnx. org/content/m 11024/latest/ View on Boundless. com

Probability and Variability Table 1 These are the four possible outcomes from flipping a coin twice. Free to share, print, make copies and changes. Get yours at www. boundless. com Open. Stax CNX. "David Lane, Binomial Distribution. May 23, 2013. " CC BY 3. 0 http: //cnx. org/content/m 11024/latest/ View on Boundless. com

Probability and Variability Coin Flip Coin flip experiments are a great way to understand the properties of binomial distributions. Free to share, print, make copies and changes. Get yours at www. boundless. com Flickr. "All sizes | Honest Abe | Flickr - Photo Sharing!. " CC BY-SA http: //www. flickr. com/photos/jeffanddayna/4081090389/sizes/m/in/photostream/ View on Boundless. com

Probability and Variability Probability Mass Function A graph of binomial probability distributions that vary according to their corresponding values for and. Free to share, print, make copies and changes. Get yours at www. boundless. com Wikipedia. "Binomial distribution. " GNU FDL http: //en. wikipedia. org/wiki/Binomial_distribution View on Boundless. com

Probability and Variability Summary of Modes This summarizes how to find the mode of a binomial distribution. Free to share, print, make copies and changes. Get yours at www. boundless. com Wikipedia. "Binomial distribution. " GNU FDL http: //en. wikipedia. org/wiki/Binomial_distribution View on Boundless. com

Probability and Variability Probability Mass Function of a Hypergeometric Distribution A random variable follows the hypergeometric distribution if its probability mass function is given by the formula shown, where: N is the population size, K is the number of success states in the population, n is the number of draws, k is the number of successes. Free to share, print, make copies and changes. Get yours at www. boundless. com Wikipedia. "Hypergeometric distribution. " GNU FDL http: //en. wikipedia. org/wiki/Hypergeometric_distribution View on Boundless. com

Probability and Variability Average Dice Value Against Number of Rolls An illustration of the convergence of sequence averages of rolls of a die to the expected value of 3. 5 as the number of rolls (trials) grows. Free to share, print, make copies and changes. Get yours at www. boundless. com Wikipedia. "Largenumbers. " CC BY-SA http: //en. wikipedia. org/wiki/File: Largenumbers. svg View on Boundless. com

Probability and Variability Attribution • Wiktionary. "random variable. " CC BY-SA 3. 0 http: //en. wiktionary. org/wiki/random_variable • Boundless Learning. "Boundless. " CC BY-SA 3. 0 http: //www. boundless. com//statistics/definition/continuous-random-variable • Boundless Learning. "Boundless. " CC BY-SA 3. 0 http: //www. boundless. com//statistics/definition/discrete-random-variable • Wikipedia. "Random variable. " CC BY-SA 3. 0 http: //en. wikipedia. org/wiki/Random_variable • Wiktionary. "probability mass function. " CC BY-SA 3. 0 http: //en. wiktionary. org/wiki/probability_mass_function • Wiktionary. "probability distribution. " CC BY-SA 3. 0 http: //en. wiktionary. org/wiki/probability_distribution • Boundless Learning. "Boundless. " CC BY-SA 3. 0 http: //www. boundless. com//statistics/definition/discrete-random-variable • Wikipedia. "Probability mass function. " CC BY-SA 3. 0 http: //en. wikipedia. org/wiki/Probability_mass_function • Wikidot. "(Discrete) Probability Distributions - Statistics. " CC BY-SA http: //statistics. wikidot. com/ch 6 • Wiktionary. "expected value. " CC BY-SA 3. 0 http: //en. wiktionary. org/wiki/expected_value • Boundless Learning. "Boundless. " CC BY-SA 3. 0 http: //www. boundless. com//statistics/definition/discrete-random-variable • Wikidot. "(Discrete) Probability Distributions - Statistics. " CC BY-SA http: //statistics. wikidot. com/ch 6 • Wikipedia. "Expected value. " CC BY-SA 3. 0 http: //en. wikipedia. org/wiki/Expected_value • Wiktionary. "probability mass function. " CC BY-SA 3. 0 http: //en. wiktionary. org/wiki/probability_mass_function • Wiktionary. "central limit theorem. " CC BY-SA 3. 0 http: //en. wiktionary. org/wiki/central_limit_theorem • Wikipedia. "Central Limit Theorem. " CC BY-SA 3. 0 http: //en. wikipedia. org/wiki/Central_Limit_Theorem • Wikipedia. "Probability mass function. " CC BY-SA 3. 0 http: //en. wikipedia. org/wiki/Probability_mass_function Free to share, print, make copies and changes. Get yours at www. boundless. com

Probability and Variability • Wikipedia. "Bernoulli trial. " CC BY-SA 3. 0 http: //en. wikipedia. org/wiki/Bernoulli_trial • Wikipedia. "Binomial distribution. " CC BY-SA 3. 0 http: //en. wikipedia. org/wiki/Binomial_distribution • Open. Stax CNX. "Susan Dean and Barbara Illowsky, Discrete Random Variables: Binomial. September 17, 2013. " CC BY 3. 0 http: //cnx. org/content/m 16820/latest/ • Open. Stax CNX. "David Lane, Binomial Distribution. September 17, 2013. " CC BY 3. 0 http: //cnx. org/content/m 11024/latest/ • Wikibooks. "Probability/Important Distributions. " CC BY-SA 3. 0 http: //en. wikibooks. org/wiki/Probability/Important_Distributions • Saylor. "MA 252: Introduction to Probability Theory « Saylor. org – Free Online Courses Built by Professors. " CC BY http: //www. saylor. org/courses/ma 252/ • Wikipedia. "Bernoulli Trial. " CC BY-SA 3. 0 http: //en. wikipedia. org/wiki/Bernoulli%20 Trial • Wikipedia. "Bernoulli distribution. " CC BY-SA 3. 0 http: //en. wikipedia. org/wiki/Bernoulli_distribution • Open. Stax CNX. "Susan Dean and Barbara Illowsky, Discrete Random Variables: Binomial. September 17, 2013. " CC BY 3. 0 http: //cnx. org/content/m 16820/latest/ • Open. Stax CNX. "Ewa Paszek, Bernoulli Trials and the Binomial Distribution. September 17, 2013. " CC BY 3. 0 http: //cnx. org/content/m 13123/latest/ • Wikipedia. "Binomial distribution. " CC BY-SA 3. 0 http: //en. wikipedia. org/wiki/Binomial_distribution • Wikibooks. "Probability/Important Distributions. " CC BY-SA 3. 0 http: //en. wikibooks. org/wiki/Probability/Important_Distributions • Open. Stax CNX. "Susan Dean and Barbara Illowsky, Discrete Random Variables: Binomial. September 17, 2013. " CC BY 3. 0 http: //cnx. org/content/m 16820/latest/ • Wiktionary. "standard deviation. " CC BY-SA 3. 0 http: //en. wiktionary. org/wiki/standard_deviation • Boundless Learning. "Boundless. " CC BY-SA 3. 0 http: //www. boundless. com//calculus/definition/mean • Wikipedia. "variance. " CC BY-SA 3. 0 http: //en. wikipedia. org/wiki/variance • Wikipedia. "Binomial distribution. " CC BY-SA 3. 0 http: //en. wikipedia. org/wiki/Binomial_distribution Free to share, print, make copies and changes. Get yours at www. boundless. com

Probability and Variability • Wikipedia. "Standard deviation. " CC BY-SA 3. 0 http: //en. wikipedia. org/wiki/Standard_deviation • Wikipedia. "Binomial distribution. " CC BY-SA 3. 0 http: //en. wikipedia. org/wiki/Binomial_distribution • Wikipedia. "Variance. " CC BY-SA 3. 0 http: //en. wikipedia. org/wiki/Variance • Open. Stax CNX. "Susan Dean and Barbara Illowsky, Discrete Random Variables: Binomial. September 17, 2013. " CC BY 3. 0 http: //cnx. org/content/m 16820/latest/ • Wikipedia. "Mean. " CC BY-SA 3. 0 http: //en. wikipedia. org/wiki/Mean • Wikipedia. "floor function. " CC BY-SA 3. 0 http: //en. wikipedia. org/wiki/floor%20 function • Wikipedia. "Mode. " CC BY-SA 3. 0 http: //en. wikipedia. org/wiki/Mode • Wikipedia. "median. " CC BY-SA 3. 0 http: //en. wikipedia. org/wiki/median • Wiktionary. "covariance. " CC BY-SA 3. 0 http: //en. wiktionary. org/wiki/covariance • Wikipedia. "Covariance. " CC BY-SA 3. 0 http: //en. wikipedia. org/wiki/Covariance • Wikipedia. "Binomial distribution. " CC BY-SA 3. 0 http: //en. wikipedia. org/wiki/Binomial_distribution • Wikipedia. "Floor function. " CC BY-SA 3. 0 http: //en. wikipedia. org/wiki/Floor_function • Wikipedia. "Median. " CC BY-SA 3. 0 http: //en. wikipedia. org/wiki/Median • Wikipedia. "Binomial distribution. " CC BY-SA 3. 0 http: //en. wikipedia. org/wiki/Binomial_distribution • Wikipedia. "Mode (statistics). " CC BY-SA 3. 0 http: //en. wikipedia. org/wiki/Mode_(statistics) • Wikipedia. "Binomial distribution. " CC BY-SA 3. 0 http: //en. wikipedia. org/wiki/Binomial_distribution • Wiktionary. "disjoint. " CC BY-SA 3. 0 http: //en. wiktionary. org/wiki/disjoint • Wiktionary. "factorial. " CC BY-SA 3. 0 http: //en. wiktionary. org/wiki/factorial • Wikipedia. "Poisson distribution. " CC BY-SA 3. 0 http: //en. wikipedia. org/wiki/Poisson%20 distribution Free to share, print, make copies and changes. Get yours at www. boundless. com

Probability and Variability • Wikipedia. "Poisson distribution. " CC BY-SA 3. 0 http: //en. wikipedia. org/wiki/Poisson_distribution • Open. Stax CNX. "Susan Dean and Barbara Illowsky, Discrete Random Variables: Poisson (optional). September 17, 2013. " CC BY 3. 0 http: //cnx. org/content/m 16829/latest/ • Wikispaces. "y 13 maths - Poisson distribution. " CC BY-SA http: //y 13 maths. wikispaces. com/Poisson+distribution • Wiktionary. "hypergeometric distribution. " CC BY-SA 3. 0 http: //en. wiktionary. org/wiki/hypergeometric_distribution • Wiktionary. "binomial distribution. " CC BY-SA 3. 0 http: //en. wiktionary. org/wiki/binomial_distribution • Wikipedia. "Bernoulli Trial. " CC BY-SA 3. 0 http: //en. wikipedia. org/wiki/Bernoulli%20 Trial • Wikipedia. "Hypergeometric distribution. " CC BY-SA 3. 0 http: //en. wikipedia. org/wiki/Hypergeometric_distribution • Open. Stax CNX. "Susan Dean and Barbara Illowsky, Discrete Random Variables: Hypergeometric (optional). September 17, 2013. " CC BY 3. 0 http: //cnx. org/content/m 16824/latest/ Free to share, print, make copies and changes. Get yours at www. boundless. com