Introduction Uniform Distribution Binomial Distribution Hypergeometric Distribution Poisson Distribution Continuous Distributions Introduction to Statistical Data Analysis Lecture 3: Probability Distributions James V. Lambers Department of Mathematics The University of Southern Mississippi James V. Lambers Statistical Data Analysis 1 / 56 Introduction Random Variables Uniform Distribution Discrete Probability Distributions Binomial Distribution Rules for Discrete Distributions Hypergeometric Distribution Mean Poisson Distribution Variance and Standard Deviation Continuous Distributions Introduction Now that we know how to compute probabilities of events, we can study the behavior of the probability across all possible outcomes of an experiment{that is, the distribution of the probability across the sample space. Our understanding of the probability distribution will eventually allow us to make inferences from the data from which the distribution arises. James V. Lambers Statistical Data Analysis 2 / 56 Introduction Random Variables Uniform Distribution Discrete Probability Distributions Binomial Distribution Rules for Discrete Distributions Hypergeometric Distribution Mean Poisson Distribution Variance and Standard Deviation Continuous Distributions Random Variables A random variable, usually denoted by a capital letter such as X , is an outcome of an experiment that has a numerical value. The value itself is usually denoted by the lower-case version of the letter used to denote the variable itself; that is, a random variable X takes on numerical values that are denoted by x. Random variables can either be continuous or discrete. A continuous random variable can assume a value equal to any real number within some interval, whereas a discrete random variable can only assume selected numerical values, such as, for example, nonnegative integers. We will study random variables of both kinds. James V. Lambers Statistical Data Analysis 3 / 56 Introduction Random Variables Uniform Distribution Discrete Probability Distributions Binomial Distribution Rules for Discrete Distributions Hypergeometric Distribution Mean Poisson Distribution Variance and Standard Deviation Continuous Distributions Discrete Probability Distributions A discrete probability distribution is a listing of all possible values of a discrete random variable, along with the probability of each value being assumed by the variable. James V. Lambers Statistical Data Analysis 4 / 56 Introduction Random Variables Uniform Distribution Discrete Probability Distributions Binomial Distribution Rules for Discrete Distributions Hypergeometric Distribution Mean Poisson Distribution Variance and Standard Deviation Continuous Distributions Example Let X be a discrete random variable whose outcomes correspond to where one finishes in a race: first, second, third, etc. If there are 10 runners in the race, then X can assume as a value any positive integer between 1 and 10. James V. Lambers Statistical Data Analysis 5 / 56 Introduction Random Variables Uniform Distribution Discrete Probability Distributions Binomial Distribution Rules for Discrete Distributions Hypergeometric Distribution Mean Poisson Distribution Variance and Standard Deviation Continuous Distributions The Distribution The probability distribution might look like the following: x P(X = x) 1 0.1 2 0.15 3 0.23 4 0.18 5 0.15 6 0.1 7 0.04 8 0.02 9 0.02 10 0.01 Note that the notation P(X = x) is used to refer to the probability that the random variable X assumes the value x. James V. Lambers Statistical Data Analysis 6 / 56 Introduction Random Variables Uniform Distribution Discrete Probability Distributions Binomial Distribution Rules for Discrete Distributions Hypergeometric Distribution Mean Poisson Distribution Variance and Standard Deviation Continuous Distributions Rules for Discrete Distributions A discrete probability distribution must follow these rules: I Each outcome must be mutually exclusive of the others; that is, we cannot have X assume two values simultaneously as a the result of an experiment. I For each outcome x, we must have 0 ≤ P(X = x) ≤ 1. I If the distribution has n possible outcomes x1; x2;:::; xn, then we must have n X P(X = xi ) = 1: i=1 James V. Lambers Statistical Data Analysis 7 / 56 Introduction Random Variables Uniform Distribution Discrete Probability Distributions Binomial Distribution Rules for Discrete Distributions Hypergeometric Distribution Mean Poisson Distribution Variance and Standard Deviation Continuous Distributions Mean For a given probability distribution, it is very helpful to know the \most likely", or expected, value that the variable will assume. This can be obtained by computing a weighted mean of the outcomes, where the probabilities serve as the weights. We therefore define the mean, or expected value, of the discrete random variable X by n X E[X ] = µ = xi P(X = xi ): i=1 James V. Lambers Statistical Data Analysis 8 / 56 Introduction Random Variables Uniform Distribution Discrete Probability Distributions Binomial Distribution Rules for Discrete Distributions Hypergeometric Distribution Mean Poisson Distribution Variance and Standard Deviation Continuous Distributions Example Consider a raffle, in which each ticket costs $5. There is one grand prize of $100, two first prizes of $50 each, and four second prizes of $25 each. If 200 tickets are sold, then the probability of winning the grand prize is 1=200 = 0:005, while the probabilities of winning first prize and second prize are 2=200 = 0:01 and 4=200 = 0:02, respectively. Then, the expected amount of winnings is E[X ] = 100(0:005) + 50(0:01) + 25(0:02) + 0(0:965) = 1:5: James V. Lambers Statistical Data Analysis 9 / 56 Introduction Random Variables Uniform Distribution Discrete Probability Distributions Binomial Distribution Rules for Discrete Distributions Hypergeometric Distribution Mean Poisson Distribution Variance and Standard Deviation Continuous Distributions Interpretation That is, a ticket holder can expect to win, on average, $1.50. However, we must account for the cost of the ticket, which applies to all participants; therefore, the expected net winnings is −$3:50. Since the expected amount is negative, the raffle is not fair to the ticket holders; if the expected value was zero, then the raffle would be considered a \fair game". James V. Lambers Statistical Data Analysis 10 / 56 Introduction Random Variables Uniform Distribution Discrete Probability Distributions Binomial Distribution Rules for Discrete Distributions Hypergeometric Distribution Mean Poisson Distribution Variance and Standard Deviation Continuous Distributions Variance and Standard Deviation Using the mean of X , we can then characterize the dispersion of the outcomes by defining the variance of X as follows: n 2 X 2 σ = (xi − µ) P(X = xi ): i=1 An equivalent formula, in terms of expected values, is σ2 = E[X 2] − E[X ]2: Note that in the first term, the values of X are squared, and then they are multiplied by the probabilities and summed, whereas in the second term, the expected value is computed first, and then squared. James V. Lambers Statistical Data Analysis 11 / 56 Introduction Uniform Distribution Binomial Distribution Hypergeometric Distribution Poisson Distribution Continuous Distributions Uniform Distribution The uniform distribution Ufa; bg is the probability distribution for a random variable X with domain fa; a + 1;:::; bg in which each value in the domain of X is equally likely to be observed It follows that the probability mass function for this distribution is 1 P(X = k) = ; n = b − a + 1; k 2 fa; a + 1;:::; bg n James V. Lambers Statistical Data Analysis 12 / 56 Introduction Uniform Distribution Binomial Distribution Hypergeometric Distribution Poisson Distribution Continuous Distributions Mean and Variance Using the above definitions of the mean and variance of a discrete random variable, it can be shown that a + b (b − a + 1)2 − 1 E[X ] = ; σ2 = 2 12 If a random variable X has the distribution Ufa; bg, we write X ∼ Ufa; bg We will use similar notation with other probability distributions, in order to indicate that a given random variable has a particular distribution James V. Lambers Statistical Data Analysis 13 / 56 Introduction Uniform Distribution Binomial Experiments Binomial Distribution The Binomial Distribution Hypergeometric Distribution The Mean and Standard Deviation Poisson Distribution Continuous Distributions Binomial Experiments Suppose that an experiment is performed n times, and it can have only two outcomes, that are classified as \success" and \failure". Each of these individual experiments is referred to as a trial. Furthermore, suppose that each trial is independent of the others, and that the probability of a trial being successful is p, where 0 < p < 1 (and therefore, the probability of failure is q = 1 − p). These trials are called Bernoulli trials. James V. Lambers Statistical Data Analysis 14 / 56 Introduction Uniform Distribution Binomial Experiments Binomial Distribution The Binomial Distribution Hypergeometric Distribution The Mean and Standard Deviation Poisson Distribution Continuous Distributions Examples Examples of Bernoulli trials are: I Testing for defective parts, in which n is the number of parts to be checked, p is the probability that a part is not defective, and k is the number of parts that are not defective. I Observing the number of correct responses on exam, in which n is the total number of questions, p is the probability of getting the correct answer on a single question, and k is