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CHS Statistics Objective: Use experimental and theoretical distributions to make judgments about the likelihood of various outcomes in uncertain situations Warm-Up Decide if the following random variable x is discrete(D) or continuous(C). 1) X represents the number of eggs a hen lays in a day. 2) X represents the amount of milk a cow produces in one day. 3) X represents the measure of voltage for a smoke-detector battery. 4) X represents the number of patrons attending a rock concert. Random Variable X Random variable - A variable, usually denoted as x, that has a single numerical value, determined by chance, for each outcome of a procedure. Probability distribution – a graph, table, or formula that gives the probability for each value of the random variable. Random Variable X A study consists of randomly selecting 14 newborn babies and counting the number of girls. If we assume that boys and girls are equally likely and we let x = the number of girls among 14 babies… What is the random variable? What are the possible values of the random variable (x)? What is the probability distribution? Probabilities of Girls x (Girls) P(x) 0 0 1 0.001 2 0.006 3 0.022 4 0.061 5 0.122 6 0.183 7 0.209 8 0.183 9 0.122 10 0.061 11 0.022 12 0.006 13 0.001 14 0 Types of Random Variables A discrete random variable has either a finite number of values or a countable number of values. A continuous random variable has infinitely many values, and those values can be associated with measurements on a continuous scale in such a ways that there are no gaps or interruptions. Usually has units Discrete Probability Distributions A Discrete probability distribution lists each possible random variable value with its corresponding probability. Requirements for a Probability Distribution: 1. All of the probabilities must be between 0 and 1. 0 ≤ P(x) ≤ 1 2. The sum of the probabilities must equal 1. ∑ P(x) = 1 Discrete Probability Distributions (cont.) The following table represents a probability distribution. What is the missing value? x 1 2 3 4 P(x) 0.16 0.22 0.28 0.2 5 Discrete Probability Distributions (cont.) Do the following tables represent discrete probability distributions? 1) 4) 2) x P(x) 0.216 5 2 0.432 3 4 x P(x) 0 3) x P(x) 0.28 1 1/2 6 0.21 2 1/4 0.288 7 0.43 3 5/4 0.064 8 0.15 4 -1 x P(x) 1 .09 2 0.36 3 0.49 4 0.06 5) P(x) = x/5, where x can be 0,1,2,3 6) P(x) = x/3, where x can be 0,1,2 Mean and Standard Deviation of a Probability Distribution Mean: 𝝁 = [𝒙 ∙ 𝑷 𝒙 ] Standard Deviation: 𝝈 = [ 𝒙𝟐 ∙ 𝑷(𝑿)] − 𝝁𝟐 Calculator: Calculate as you would for a weighted mean or frequency distribution: Very important! Stat Edit L1 = x values L2 = P(x) values Stat Calc 1: Variable Stats L1, L2 Mean and Standard Deviation of a Probability Distribution (cont.) Calculate the mean and standard deviation of the following probability distributions: 1) Let x represent the # of games required to complete the World Series: 2) Let x represent the # dogs per household: X = # of Dogs Households 0 1491 0.253 1 425 6 0.217 2 168 7 0.410 3 48 x P(x) 4 0.480 5 Expected Value The expected value of a discrete random variable represents the average value of the outcomes, thus is the same as the mean of the distribution. 𝑬=𝝁= [𝒙 ∙ 𝑷 𝒙 ] Expected Value Consider the numbers game, often called “Pick Three” started many years ago by organized crime groups and now run legally by many governments. To play, you place a bet that the three-digit number of your choice will be the winning number selected. The typical winning payoff is 499 to 1, meaning for every $1 bet, you can expect to win $500. This leaves you with a net profit of $499. Suppose that you bet $1 on the number 327. What is your expected value of gain or loss? What does this mean? Event Win Lose x P(x) Assignment pp. 190 # 2 – 14 Even, 18 – 22 Even