Intensive Algebra 2 Name______________________ Probability Functions & Discrete Random Variables May 17, 2011 Introduction to Discrete Random Variables A discrete random variable is one in which we can produce a countable number of outcomes, and is usually associated with the counting process (i.e. the number of plants that will bloom, or the number of defective items in a shipment, or the number of flips that show “tails”). A random variable, X (random variables are usually denoted by capital letters), can take on exactly n values, each of which corresponds to one and only one of the events in the sample space. Notation for this concept looks like this: P( X x) p , which is translated as “the probability that the random variable X = x is p.” We can describe a discrete random variable by making use of its probability distribution, that is, by showing the values of the random variable and the probabilities associated with each of its values. The distribution can be displayed in any of the following formats: 1. In a table 2. Graphically (with the probability value on the vertical axis and the values of the random variable on the horizontal axis) 3. As a function (a formula that can be used to determine the probability values) Example: Given an experiment of tossing a coin three times, use each of the probability distribution representations to display the results of this experiment. First, let the random variable X denote the number of heads observed in three tosses of a coin. 1. In a table: x 0 1 P(X = x) 2. Graphically: P(X = x) x 3. As a function: P(X = x) = for x = 0, 1, 2, 3 2 3 Intensive Algebra 2 Platt/Knaus Exercises: 1. Two friends, Kirsty and Bridget, independently applied for two different jobs. The chances that Kirsty is successful is 0.8 and the chances that Bridget is successful is 0.75. a. If X denotes the number of successful applications between the two friends, find the probability distribution of X and display it both in a table and graphically. b. Find the probability that if one is successful, it is Kirsty. 2. A bag contains 5 white cubes and 4 red cubes. Two cubes are selected in such a way that the first cube drawn is not replaced before the next cube is drawn. Find the probability distribution (table & graph) of X, where X denotes the number of white cubes selected from the bag. (hint: you may also want to make a tree!) What is the probability that both cubes are red? Intensive Algebra 2 Platt/Knaus 3. Consider the random variable X with probability function defined by P ( X 0) , P( X 1) 2 , and P ( X 2) 3 Determine the value of alpha. 4. The probability distribution of the random variable X is represented by the function: P( X x) k x for x = 1,2,3,4,5,6 a. Find the value of k. b. Find P (3 X 5) 5. A discrete random variable X has a probability distribution defined by the function: x 2 3 P( X x) 4 Cx 5 5 4 x where x = 0,1,2,3,4 a. Display this distribution in tabular (in a table) and graphical forms. b. Find P( X 2) and P(1 X 3) . Intensive Algebra 2 Platt/Knaus HW Exercises: 1. 2. 3. Intensive Algebra 2 Platt/Knaus 4. 5. Intensive Algebra 2 Platt/Knaus 6. 7. 8.