Discrete Random Variables: Probability Distribution Function (PDF) for a Discrete Random Variable 1.11 2008/05/16 14:38:08 GMT-5 2009/01/25 17:59:14.856 US/Central Susan Dean deansusan@deanza.edu Barbara Illowsky illowskybarbara@deanza.edu Susan Dean deansusan@deanza.edu Barbara Illowsky illowskybarbara@deanza.edu Connexions cnx@cnx.org discrete distribution elementary function PDF probability random statistics variable This module introduces the Probability Distribution Function (PDF) and its characteristics. A discrete probability distribution function has two characteristics: Each probability is between 0 and 1, inclusive. The sum of the probabilities is 1. P(X) is the notation used to represent a discrete probability distribution function. A child psychologist is interested in the number of times a newborn baby's crying wakes its mother after midnight. For a random sample of 50 mothers, the following information was obtained. Let X = the number of times a newborn wakes its mother after midnight. For this example, x = 0, 1, 2, 3, 4, 5. P(X = x) = probability that X takes on a value x. x P(X = x) 0 P(X=0)=250 1 P(X=1)=1150 2 P(X=2)=2350 3 P(X=3)=950 4 P(X=4)=450 5 P(X=5)=150 X takes on the values 0, 1, 2, 3, 4, 5. This is a discrete PDF because Each P(X = x) is between 0 and 1, inclusive. The sum of the probabilities is 1, that is, 2 50 + 11 50 + 23 50 + 9 50 + 4 50 + 1 50 = 1 Suppose Nancy has classes 3 days a week. She attends classes 3 days a week 80% of the time, 2 days 15% of the time, 1 day 4% of the time, and no days 1% of the time. Let X = the number of days Nancy ____________________ . Let X = the number of days Nancy attends class per week. X takes on what values? 0, 1, 2, and 3 Construct a probability distribution table (called a PDF table) like the one in the previous example. The table should have two columns labeled x and P(X = x). What does the P(X = x) column sum to? x P(X = x) 0 0.01 1 0.04 2 0.15 3 0.80 Probability Distribution Function (PDF) A mathematical description of a discrete random variable (RV), given either in the form of an equation (formula) , or in the form of a table listing all the possible outcomes of an experiment and the probability associated with each outcome. A biased coin with probability 0.7 for a head (in one toss of the coin) is tossed 5 times. We are interested in the number of heads (the RV X = the number of heads). X is Binomial, so X ∼ B ( 5 , 0 . 7 ) and P ( X = x ) = 5 x . 7 x . 3 5 − x or in the form of the table: x P ( X = x ) 0 0.0024 1 0.0284 2 0.1323 3 0.3087 4 0.3602 5 0.1681