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1 Discrete and Continuous Random Variables: A __________ is a quantity whose value changes. A __________ __________ is a variable whose value is obtained by __________. number of students present number of red marbles in a jar number of heads when flipping three coins students’ grade level Examples: A __________ __________ is a variable whose value is obtained by __________. height of students in class weight of students in class time it takes to get to school distance traveled between classes Examples: A __________ __________ is a variable whose value is a numerical outcome of a random phenomenon. ▪ A random variable is denoted with a ____________________ ▪ The __________ __________ of a random variable X tells what the possible values of X are and how probabilities are assigned to those values ▪ A random variable can be __________ or __________ A __________ __________ __________ X has a countable number of possible values. Example: Let X represent the sum of two dice. Then the probability distribution of X is as follows: X 2 3 4 5 6 7 8 9 10 11 12 P(X) 1 36 2 36 3 36 4 36 5 36 6 36 5 36 4 36 3 36 2 36 1 36 To graph the probability distribution of a __________ __________ __________, construct a ____________________. 2 Probability Distribution of X Probability 0.20 0.15 0.10 0.05 0.00 2 3 4 5 6 7 8 9 10 11 12 Outcome A __________ __________ __________ X takes all values in a given interval of numbers. ▪ The probability distribution of a continuous random variable is shown by a __________ __________ . ▪ The probability that X is between an interval of numbers is the __________ under the density curve between the interval endpoints Means and Variances of Random Variables: The __________ of a discrete random variable, X, is its ____________________. Each value of X is weighted by its probability. To find the mean of X, __________ each value of X by its probability, then __________ all the products. μ X = x1 p1 + x2 p2 + ⋅⋅⋅ + xk pk = ∑ xi pi The mean of a random variable X is called the __________ __________ of X. Law of Large Numbers: As the number of observations increases, the mean of the ____________________, x , approaches the mean of the __________, μ . The more __________ in the outcomes, the more trials are needed to ensure that close to μ . x is 3 Rules for Means: If X is a random variable and a and b are fixed numbers, then If X and Y are random variables, then Example: Suppose the equation Y = 20 + 100X converts a PSAT math score, X, into an SAT math score, Y. Suppose the average PSAT math score is 48. What is the average SAT math score? μ X = 48 μ a +bX = a + bμ X μ 20+100 X = 20 + 100 μ X = 20 + 100 ( 48 ) = 500 Example: Let Let μ X = 625 μY = 590 represent the average SAT math score. represent the average SAT verbal score. μ X +Y = μ X + μY represents the average combined SAT score. Then μ X +Y = μ X + μY = 625 + 590 = 1215 is the average combined total SAT score. The Variance of a Discrete Random Variable: If X is a discrete random variable with mean μ , then the __________ of X is σ 2 = ( x1 − μ X ) p1 + ( x2 − μ X ) p2 + ⋅⋅⋅ + ( xk − μ X ) pk 2 2 2 X = ∑ ( xi − μ X ) pi 2 The standard deviation __________ is the __________ of the __________. 4 Rules for Variances: If X is a random variable and a and b are fixed numbers, then If X and Y are independent random variables, then Example: Suppose the equation Y = 20 + 100X converts a PSAT math score, X, into an SAT math score, Y. Suppose the standard deviation for the PSAT math score is 1.5 points. What is the standard deviation for the SAT math score? σ X2 = (1.5 ) = 2.25 2 σ a2+bX = b 2σ X2 σ 202 +100 X = (100 ) σ X2 2 = (100 ) ( 2.25 ) 2 = 22, 500 σ X2 = 150 Suppose the standard deviation for the SAT math score is 150 points, and the standard deviation for the SAT verbal score is 165 points. What is the standard deviation for the combined SAT score? *** Because the SAT math score and SAT verbal score are not __________, the rule for adding __________ does not apply!