Section 8.1 Distributions of Random Variables Random Variable A random variable is a rule that assigns a number to each outcome of a chance experiment. There are three types of random variables: 1. Finite Discrete: The random variable has a finite number, n, of values it can take on, and the random variable can only assume whole number values, like {0, 1, 2,..., n}. 2. Infinite Discrete: The random variable has an infinite number of values it can take on, and the random variable can only assume whole number values, like {0, 1, 2, 3,...}. 3. Continuous The random variable has an infinite number of values it can take on, and the random variable can assume fractional number values, like {1/2, 2/3, 3/5,...}. 1. Consider the following. X = The number of times a die is thrown until a 2 appears Give the range of values that the random variable X may assume. Classify the random variable. 2. Consider the following. X = The number of hours a child watches television on a given day Give the range of values that the random variable X may assume. Classify the random variable.
3. Cards are selected one at a time without replacement from a well-shuffled deck of 52 cards until an ace is drawn. Let X denote the random variable that gives the number of cards drawn. What values may X assume? 4. Determine the possible values of the given random variable and indicate as your answer whether the random variable is finite discrete, infinite discrete, or continuous. A marble is drawn at random and then replaced from a box of 7 red and 6 green marbles. Let the random variable X be the number of draws until a a red marble is picked. What are the possible values of X? Classify X. Probability Distribution for a Random Variable X If X = {x 1, x 2,, x n } is a random variable with the given set of values, then the probability distribution for the random variable is a table where the entries in the first row are all the possible values X can assume (x 1, x 2,, x n ) and the entries in the second row are all their corresponding probabilities (P (X = x 1 ), P (X = x 2 ),..., P (X = x n )). x x 1 x 2 x n P (X = x) P (x 1 ) P (x 2 ) P (x n ) 2 Fall 2016, Maya Johnson
5. The probability distribution of the random variable X is shown in the accompanying table. x 10 5 0 5 10 15 20 P (X = x).20.10.25.15.05.15.10 Find the following. (a) P (X = 10) (b) P (X 5) (c) P ( 5 X 5) (d) P (X 20) (e) P (X < 5) (f) P (X = 2) 6. A survey was conducted by the Public Housing Authority in a certain community among 1000 families to determine the distribution of families by size. The results are given below. Family Size 2 3 4 5 6 7 8 Frequency of Occurrence 300 209 207 80 69 12 123 Find the probability distribution of the random variable X, where X denotes the number of persons in a randomly chosen family. (Give answers as fractions.) 3 Fall 2016, Maya Johnson
Family Size 2 3 4 5 6 7 8 P (X = x) 7. Two cards are drawn from a well-shuffled deck of 52 playing cards. Let X denote the number of aces drawn. Find the probability distribution of the random variable X. (Round answer to three decimal places.) 8. Let X denote the random variable that gives the sum of the faces that fall uppermost when two fair dice are rolled. Find P (X = 7). (Round answer to two decimal places.) 9. A box has 5 yellow, 7 gray, and 3 black marbles. Three marbles are drawn at the same time (i.e. without replacement) from the box. Let X be the number of gray marbles drawn. Find the following. (Round answers to three decimal places.) (a) P (X = 2) 4 Fall 2016, Maya Johnson
(b) P (X 2) Histograms A histogram is a graphical representation of a probability distribution of a random variable X. The horizontal axis represents all the possible values the random variable X may assume, while the vertical axis represents their corresponding probabilities. 10. An examination consisting of ten true-or-false questions was taken by a class of 100 students. The probability distribution of the random variable X, where X denotes the number of questions answered correctly by a randomly chosen student, is represented by the accompanying histogram. The rectangle with base centered on the number 8 is missing. What should be the height of this rectangle? 5 Fall 2016, Maya Johnson