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Mean of a Random Variable Suppose that X is a discrete random variable whose distribution is as follows : Value of X x1 x2 x3 ... xk Probability p1 p2 p3 ... pk To find the mean of X, multiply each possible value by its probability, then add all the products. X = x1p1 + x2p2 + x3p3 + … + xkpk = xipi Mean of a Random Variable Example : Nelson is trying to get Homer to play a game of chance. This game costs $2 to play. The game is to flip a coin three times, and for each head that appears, Homer will win $1. Here is the probability distribution : Number of Heads Probability 0 1 2 3 .125 .375 .375 .125 What is the expected payoff of this game? The mean! Mean of a Random Variable Number of Heads Probability 0 1 2 3 .125 .375 .375 .125 X = (0)(.125) + (1)(.375) + (2)(.375) + (3)(.125) = 1.5 In this game, Homer would expect to win $1.50 So, should Homer play this game ? No. Rules for Means 1) If X is a random variable and a and b are fixed numbers, then : a+bX = a + bX Example : What if Homer tries Nelsons game, but he decides to award $10 for every head that Nelson flips. Number of Heads Probability 0 10 20 30 .125 .375 .375 .125 X = (0)(.125) + (10)(.375) + (20)(.375) + (30)(.125) = 15 Rules for Means 1) If X is a random variable and a and b are fixed numbers, then : a+bX = a + bX Example : What if Homer tries Nelsons game, but he decides to award $10 for every head that Nelson flips. Number of Heads Probability 0 10 20 30 .125 .375 .375 .125 Notice that this is just a linear transformation with b = 10 and a = 0. 10X = a + bX = 0 + 10X = (10)(1.5) = 15 (Same as before) Rules for Means 2) If X and Y are random variables, then X+Y = X + Y Example : Let X be a random variable with X = 12 Let Y be a random variable with Y = 18 What is the mean of X + Y ? X+Y = X + Y = 12 + 18 = 30 Variance of a Discrete Random Variable Suppose that X is a discrete random variable whose distribution is : Value of X x1 x2 x3 ... xk Probability p1 p2 p3 ... pk and that X is the mean of X. The variance of X is : X2 = (x1 - X)2 p1 + (x2 - X)2 p2 + ... + (xk - X)2 pk = (xi - X)2 pi The standard deviation the variance. X of X is the square root of Variance of a Discrete Random Variable Example : Find the variance of Nelson’s game. Number of Heads Probability 0 1 2 3 .125 .375 .375 .125 Recall the mean was 1.5 X2 = (x1 - X)2 p1 + (x2 - X)2 p2 + ... + (xk - X)2 pk = (0 - 1.5)2 (.125) + (1 - 1.5)2 (.375) + (2 - 1.5)2 (.375) + (3 - 1.5)2 (.125) = .28125 + .09375 + .09375 + .28125 = 0.75 X = 0.75 = .8660254 Rules for Variances 1) If X is a random variable and a and b are fixed numbers, then : a2+ bX = b2 2 X 2) If X and Y are independent random variables, then X2+ Y = X2 + Y2 X2- Y = X2 + Y2 Rules for Variances Example : Mike’s golf score varies from round to round. X = 88 X = 8 Tiger’s golf score vary from round to round. Y = 92 Y = 9 Q: What is the expected difference between our two rounds ? X-Y = X - Y = 88 - 92 = -4 Rules for Variances Example : Mike’s golf score varies from round to round. X = 88 X = 8 Tiger’s golf score vary from round to round. Y = 92 Y = 9 Q: What is the variance of the differences ? X2- Y = X2 + Y2 = 64 + 81 = 145 X-Y = 145 = 12.041594 The Law of Large Numbers • Draw independent observations at random from any population with finite mean • Decide how accurately you want to estimate • As the number of observations increases, the mean of the observed values eventually approaches the mean of the population. Example: M & M’s • Brown 30% • Red 20% • Yellow 20% • Blue 10% • Green 10% • Orange 10% M & M’s • Brown 30% • Red 20% • Yellow 20% • Blue 10% • Green 10% • Orange 10% Law of Large Numbers During an experiment, as the number of experiments grows larger, then the observed mean will grow closer and closer to the actual mean. M & M’s • Brown 30% • Red 20% • Yellow 20% Red Totals 0 Brown 0 Yellow Blue 0 0 Green Orange 0 0 • Blue 10% • Green 10% • Orange 10% Totals Red Brown Yellow Blue Green Orange #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! 20 30 20 10 10 10 Homework 50, 51, 54, 55, 56, 58, 60, 61, 62, 66, 69