Math 141: Business Mathematics I Fall 2015 §8.1 Distributions of Random Variables Instructor: Yeong-Chyuan Chung Random Variables A random variable is a rule that assigns a number to each outcome of an experiment. We may also think of a random variable as a function that gives us a number whenever we input an outcome of an experiment. We usually use uppercase letters for random variables. Example. If the experiment involves tossing a coin three times, then we can consider the random variable X that represents the number of times we get tails. In this case, the possible values of X are 1, 2, and 3. Example. If the experiment involves rolling a die repeatedly until a 5 occurs, then we can consider the random variable Y that represents the number of times we roll the die. In this case, the possible values of Y are 1, 2, 3, . . . Example. Suppose that a flashlight is left on until the battery runs out in an experiment to determine the battery life. We can consider the random variable Z that represents the battery life. In this case, the possible values of Z are all the nonnegative numbers. In these three examples, we see three different types of random variables distinguished by the values they can attain. Type of random variable Finite discrete Infinite discrete Continuous Property Can take on only finitely many values Can take on infinitely many values, which may be listed as a sequence Can take on infinitely many values, which make up an interval of real numbers Example. For each of the following random variables, give the values, or range of values, that the random variable can take on, and classify the random variable as finite discrete, infinite discrete, or continuous. (a) The number of defective iPads in a sample of eight iPads. 1 §8.1 Distributions of Random Variables 2 (b) The height of a randomly chosen college student. (c) The number of times a card is picked from a deck of 52 cards with replacement until an ace is picked. (d) The number of times a card is picked from a deck of 52 cards without replacement until an ace is picked. Probability Distributions of (finite discrete) Random Variables In §7.2, we considered probability distributions associated to experiments. There, we assigned probabilities to each outcome of an experiment and tabulated these probabilities to obtain the probability distribution. Now that we have introduced the notion of a random variable, we will consider the probability distribution associated with the random variable rather than one associated with the outcomes of the experiment. This will be a table containing the list of all possible values the random variable can attain and the associated probabilities. If X is a random variable, and x is one of the possible values of X, then we write P (X = x) to mean the probability that X takes the value x. The probability distribution of X will look like x P (X = x) x1 p1 x2 p2 x3 p3 ··· ··· xn pn where x1 , x2 , . . . , xn are all the possible values of X, and p1 , p2 , . . . , pn are the respective probabilities. We may also draw the table vertically instead of horizontally. Just as before, each individual probability appearing in the table is a number between 0 and 1, inclusive, and the sum of all of them is 1. §8.1 Distributions of Random Variables 3 Example (Exercise 14 in the text). Determine whether the following table gives the probability distribution of the random variable X. Explain your answer. x P (X = x) −2 0.2 −1 0 1 2 0.1 0.3 0.2 0.1 Example (Exercise 18 in the text). Find conditions on the numbers a and b such that the table gives the probability distribution of the random variable X. x −1 0 1 2 4 P (X = x) 0.3 a 0.2 0.2 b 5 0.1 Histograms Histograms can be used to represent probability distributions graphically. These are diagrams in which the possible values of the random variable are marked on the horizontal axis, and a rectangle is drawn above each value with height equal to the probability associated with that value. If the rectangles are each drawn with width 1, then the probability associated with each value is the same as the area of the corresponding rectangle. Example (Exercise 20 in the text). The probability distribution of the random variable X is shown below. x P (X = x) −5 −3 −2 .17 .13 .33 0 2 3 .16 .11 .10 Draw a histogram representing this probability distribution. Find P (−2 < X ≤ 2) and P (X = 1). §8.1 Distributions of Random Variables 4 Example (Exercise 30 in the text). The following table gives the number of U.S. households within each of four income groups in which someone owns a smartphone. Income ($) Number of owners (in thousands) <30,000 11,817 30,000-49,999 8,475 50,000-74,999 7,609 75,000+ 32,057 Let X denote the random variable that takes on the values 1, 2, 3, and 4, corresponding to the income groups <$30,000, $30,000-$49,999, $50,000-$74,999, and $75,000+, respectively. (a) Find the probability distribution associated with these data. (b) Draw a histogram representing the probability distribution. (c) If a smartphone owner in one of these households is chosen at random, what is the probability that his or her household income is between $30,000 and $74,999?