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Mean of a Random Variable

Mean of a Random Variable. x 1. x 2. x 3. x k. Value of X. p 1. p 2. p 3. p k. Probability. Suppose that X is a discrete random variable whose distribution is as follows :. To find the mean of X, multiply each possible value by its probability, then add all the products.

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Mean of a Random Variable

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  1. Mean of a Random Variable x1 x2 x3 ... xk Value of X p1 p2 p3 ... pk Probability Suppose that X is a discrete random variable whose distribution is as follows : To find the mean of X, multiply each possible value by its probability, then add all the products. X = x1p1 + x2p2 + x3p3 + … + xkpk =  xipi

  2. Mean of a Random Variable Number of Heads 0 1 2 3 Probability Nelson is trying to get Homer to play a game of chance. Example : 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 : .125 .375 .375 .125 What is the expected payoff of this game? The mean!

  3. Mean of a Random Variable .125 .375 .375 .125 Number of Heads 0 1 2 3 Probability 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.

  4. Rules for Means .125 .375 .375 .125 Number of Heads 0 10 20 30 Probability 1) If X is a random variable and a and b are fixed numbers, then : a+bX = a + bX Example : What if Homer tries Nelsons game, but he decides to award $10 for every head that Nelson flips. X = (0)(.125) + (10)(.375) + (20)(.375) + (30)(.125) = 15

  5. Rules for Means .125 .375 .375 .125 Number of Heads 0 10 20 30 Probability 1) If X is a random variable and a and b are fixed numbers, then : a+bX = a + bX Example : What if Homer tries Nelsons game, but he decides to award $10 for every head that Nelson flips. Notice that this is just a linear transformation with b = 10 and a = 0. 10X = a + bX = 0 + 10X = (10)(1.5) = 15 (Same as before)

  6. 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

  7. Variance of a Discrete Random Variable x1 x2 x3 ... xk Value of X p1 p2 p3 ... pk Probability  2 = (x1 - X)2 p1 + (x2 - X)2 p2 + + (xk - X)2 pk ... X Suppose that X is a discrete random variable whose distribution is : and that X is the mean of X. The variance of X is : = (xi - X)2 pi The standard deviation X of X is the square root of the variance.

  8. Variance of a Discrete Random Variable .125 .375 .375 .125 Number of Heads 0 1 2 3 Probability  2 = (x1 - X)2 p1 + (x2 - X)2 p2 + + (xk - X)2 pk ... X 0.75 Example : Find the variance of Nelson’s game. Recall the mean was 1.5 = (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 = = .8660254

  9. Rules for Variances         2 2 2 2 2 2 2 2 = = + + X X X + Y Y a + bX X Y X - Y = b2 1) If X is a random variable and a and b are fixed numbers, then : 2) If X and Y are independent random variables, then

  10. 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

  11. Rules for Variances    2 2 2 = + X Y X - Y 145 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 ? = 64 + 81 = 145 X-Y = = 12.041594

  12. 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.

  13. Example:

  14. M & M’s • Brown 30% • Red 20% • Yellow 20% • Blue 10% • Green 10% • Orange 10%

  15. 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.

  16. Blue 10% • Green 10% • Orange 10% • Brown 30% • Red 20% • Yellow 20% M & M’s

  17. Homework 50, 51, 54, 55, 56, 58, 60, 61, 62, 66, 69

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