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Unformatted text preview: Chapter 4: RVs and PDFs (will be updated until end of this chapter lectures) Sections 4.1,4.2,4.3,4.5,4.6, 1.6 (data collection), 4.7, 4.10,4.11 Section 4.1: Random Variables A Random variable (RV) is a rule or function or process which assigns numerical values to the outcomes of a random experiment. We denote a RV with capital letter. A coin is tossed once. Do the outcomes of this experiment is a RV? A coin is tossed twice. Do the outcomes of this experiment is a RV? A Die is rolled. Do the outcomes of this experiment is a RV? Two dice are rolled. Does the total of this experiment is a RV? You wait for a Bus, whose frequency is every 10mins. Is the waiting time a RV? There are 2-types of RVs: 1. Discrete RV (DRV): The values of this RV are discrete numerical values 2. Continuous RV (CRV): The values of this RV are continuous numerical values Section 4.2: PDF of a DRV The Probability Distribution Function (pdf) of a discrete random variable is a table or a graph or a function which provides information about different values this RV takes and probabilities of these values occur. However the PDF of a DRV must satisfy the following requirements: Suppose X takes discrete numerical values k x x x , , , 2 1 and suppose the probabilities assigned to these values k k p x X P p x X P p x X P = = = = = = ) ( , , ) ( , ) ( 2 2 1 1 respectively, then these probabilities must satisfy the following rules: 1. 1 ) ( j x P for all values k j , , 2 , 1 = 2. 1 ) ( ) ( ) ( ) ( 2 1 1 = + + + = = k n j j x P x P x P x P Example 1: The number of Heads in two tosses has the distribution table: X 1 2 P Does this table represent a PDF? Example 2: The number of tails in three tosses has the distribution table: X 1 2 3 P 1/8 3/8 3/8 1/8 Does this table represent a PDF? Example 3: Vegas odds on which city (X) will win the World Series: XNY PH SF TB TX P 3/7 4/6 1/9 1/9 1/9 Does this table represent a PDF? Example 4: Early predictions for NCAA basketball champion by conference CONFERENCE TEAM S Prob Big East 11 0.2 Big Ten 7 0.25 Big 12 6 0.3 ACC 5 0.075 SEC 4 0.075 Pac-10 4 0.05 Mountain West 3 0.01 Atlantic 10 3 0.01 West Coast 2 0.01 Colonial 2 0.01 Other 21 0.01 Does this table represent a PDF? There are advantages with Random Variable instead of having a general concept of outcomes of an experiment.-You can define cumulative probabilities for a RV. There are 3 types them. Less than (at most), greater than (at least), in between (interval). P( ) x X := prob. of X takes values less than or equal to a number x. P( ) x X := prob. of X takes values greater than or equal to a number x. P( ) 2 1 x X x := prob. of X takes values in between numbers 1 x and 2 x including them....
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- Spring '10