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6-2 Discrete Probability Distributions Chapter Contents 6.1 Discrete Distributions 6.2 Uniform Distribution 6.3 Bernoulli Distribution 6.4 Binomial Distribution 6.5 Poisson Distribution 6.6 Hypergeometric Distribution 6.7Geometric Distribution (Optional) 6.7 Geometric Distribution (Optional) 6.8Transformations of Random Variables (Optional) 6.8 Transformations of Random Variables (Optional) Chapter 6

6-3 Chapter Learning Objectives LO6-1: Define a discrete random variable. LO6-2: Solve problems using expected value and variance. LO6-3: Define probability distribution, PDF, and CDF. LO6-4: Know the mean and variance of a uniform discrete model. LO6-5: Find binomial probabilities using tables, formulas, or Excel. LO6-6: Find Poisson probabilities using tables, formulas, or Excel. LO6-7: Use the Poisson approximation to the binomial (optional). LO6-8: Find hypergeometric probabilities using Excel. LO6-9: Calculate geometric probabilities (optional). LO6-10: Select an appropriate discrete model from problem context. LO6-11: Apply rules for transformations of random variables (optional). Chapter 6 Discrete Probability Distributions

6-4 random variableA random variable is a function or rule that assigns a numerical value to each outcome in the sample space of a random experiment. discrete random variableA discrete random variable has a countable number of distinct values Random Variables Random Variables 6.1 Discrete Distributions Chapter 6 LO6-1: Define a discrete random variable. Probability Distributions A discrete probability distribution assigns a probability to each value of a discrete random variable X.A discrete probability distribution assigns a probability to each value of a discrete random variable X. To be a valid probability distribution, each of the following must be satisfied.To be a valid probability distribution, each of the following must be satisfied. LO6-1

6-5 The expected value E(X) of a discrete random variable is the sum of all X-values weighted by their respective probabilities.The expected value E(X) of a discrete random variable is the sum of all X-values weighted by their respective probabilities. The E(X) is a measure of central tendency.The E(X) is a measure of central tendency. Expected Value Chapter 6 LO6-2: Solve problems using expected value and variance. LO Discrete Distributions

6-6 If there are n distinct values of X, then the variance of a discrete random variable is:If there are n distinct values of X, then the variance of a discrete random variable is: The variance is a weighted average of the dispersion about the mean and is denoted either as  2 or V(X).The variance is a weighted average of the dispersion about the mean and is denoted either as  2 or V(X). The standard deviation is the square root of the variance and is denoted .The standard deviation is the square root of the variance and is denoted . Variance and Standard Deviation Chapter Discrete Distributions LO6-2

6-7 A probability distribution function (PDF) is a mathematical function that shows the probability of each X-value.A probability distribution function (PDF) is a mathematical function that shows the probability of each X-value. A cumulative distribution function (CDF) is a mathematical function that shows the cumulative sum of probabilities, adding from the smallest to the largest X-value, gradually approaching unity.A cumulative distribution function (CDF) is a mathematical function that shows the cumulative sum of probabilities, adding from the smallest to the largest X-value, gradually approaching unity. What is a PDF or CDF? Chapter 6 LO6-3: Define probability distribution, PDF, and CDF. 6.1 Discrete Distributions LO6-3

6-8 Illustrative PDF (Probability Density Function) Cumulative CDF (Cumulative Density Function) Consider the following illustrative histograms: The equations for these functions depend on the parameter(s) of the distribution. What is a PDF or CDF? Chapter 6 PDF = P(X = x) CDF = P(X ≤ x) 6.1 Discrete Distributions LO6-3

6-9 Characteristics of the Uniform Discrete Distribution The uniform distribution describes a random variable with a finite number of integer values from a to b (the only two parameters).The uniform distribution describes a random variable with a finite number of integer values from a to b (the only two parameters). Each value of the random variable is equally likely to occur.Each value of the random variable is equally likely to occur. Chapter Uniform Distribution LO6-4: Know the mean and variance of a uniform discrete model. LO6-4

6-10 A random experiment with only 2 outcomes is a Bernoulli experiment. One outcome is arbitrarily labeled a “success” (denoted X = 1) and the other a “failure” (denoted X = 0).  is the P(success), 1 –  is the P(failure). Note that P(0) + P(1) = (1 –  ) +  = 1 and 0 ≤  ≤ 1. “Success” is usually defined as the less likely outcome so that  <.5 for convenience. Bernoulli Experiments Chapter Bernoulli Distribution The expected value (mean) and variance of a Bernoulli experiment is calculated as: E(X) =  and V(X) =  (1 -  )

6-11 The binomial distribution arises when a Bernoulli experiment is repeated n times.The binomial distribution arises when a Bernoulli experiment is repeated n times. Characteristics of the Binomial Distribution Chapter Binomial Distribution LO6-5: Find binomial probabilities using tables, formulas, or Excel. LO6-5

6-12 Called the model of arrivals, most Poisson applications model arrivals per unit of time.Called the model of arrivals, most Poisson applications model arrivals per unit of time. Chapter Poisson Distribution LO6-6: Find Poisson probabilities using tables, formulas, or Excel. Characteristics of the Poisson Distribution LO6-6

6-13 The hypergeometric distribution is similar to the binomial distribution. However, unlike the binomial, sampling is without replacement from a finite population of N items. The hypergeometric distribution may be skewed right or left and is symmetric only if the proportion of successes in the population is 50%. Characteristics of the Hypergeometric Distribution. 6.6 Hypergeometric Distribution LO6-8: Find hypergeometric probabilities using Excel. LO6-8

6-14 Both the binomial and hyper geometric involve samples of size n and treat X as the number of successes. The binomial samples with replacement while the hyper geometric samples without replacement. Binomial Approximation to the Hypergeometric 6.6 Hypergeometric Distribution LO6-8

6-15 The geometric distribution describes the number of Bernoulli trials until the first success. X is the number of trials until the first success. X ranges from {1, 2,...} since we must have at least one trial to obtain the first success. However, the number of trials is not fixed.  is the constant probability of a success on each trial Characteristics of the Geometric Distribution 6.7 Geometric Distribution LO6-9: Calculate Geometric probabilities. LO6-9

6-16 A linear transformation of a random variable X is performed by adding a constant or multiplying by a constant.A linear transformation of a random variable X is performed by adding a constant or multiplying by a constant. Linear Transformations 6.8 Transformation of Random Variables LO6-11: Apply rules for transformations of random variables. LO6-11