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7.2 MEANS AND VARIANCES OF RANDOM VARIABLES (Pages 385 - 404)

7.2 MEANS AND VARIANCES OF RANDOM VARIABLES (Pages 385 - 404). "It is the nature of every man to err, but only the fool perseveres in error." Cicero (Roman statesman, 106 - 46 B.C.). OVERVIEW:. If X is a discrete random variable with possible values x i having probabilities p i , then

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7.2 MEANS AND VARIANCES OF RANDOM VARIABLES (Pages 385 - 404)

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  1. 7.2 MEANS AND VARIANCES OF RANDOM VARIABLES (Pages 385 - 404) "It is the nature of every man to err, but only the fool perseveres in error." Cicero (Roman statesman, 106 - 46 B.C.)

  2. OVERVIEW: • If X is a discreterandomvariable with possible values xi having probabilities pi, then • mean of x values • = m = x1p1 + x2p2 + ... + xkpk • = sum(xipi), and • variance of x values • = s2= (x1-m)2p1 + (x2-m)2p2 + ... + (xk-m)2pk • = sum[(xi-m)2pi] • This section provides illustrations using these formulas.

  3. Example: • Consider a random variable x that assumes the values 1,2,3 with respective probabilities 60%, 30%, 10%. • The following table illustrates use of the formulas in the OVERVIEW.

  4. The formulas shown in the OVERVIEW are consistent with previously established formulas for mean, standard deviation, and variance. • To illustrate this, note that in the previous example, one would "expect" this distribution of the random variable in ten trials: {1,1,1,1,1,1,2,2,2,3}. • If you place this set in list L1 and calculate 1-Var Stats, you will find that the mean is 1.5 and the standard deviation is 0.6708203932. • Squaring the standard deviation to obtain variance yields 0.4499999999, or 0.45.

  5. Roulette Example • As an example, there are 18 black slots, 18 red slots, and 2 green slots in American roulette. If you bet $1 on one number (such as black 17, the favorite of James Bond), you have a probability of 1/38 of winning $35. The probability you will lose your dollar is 37/38. Your expectation is ($35)(1/38) - ($1)(37/38) = -$.0526. • Roulette is a fun game, and it is certainly possible for you to win some money if you play for a short period of time. However, math power easily illustrates that you can't expect to make a living playing this game in a casino. • Of course, it has to be admitted that James Bond always seems to do unusually well in casinos. A good gambling strategy would be to follow him around and put your money on whatever he bets on!

  6. Law of Large Numbers: This says that the actual mean of many trials gets close to the distribution mean as more trials are made. • Example: When you flip a coin numerous times, you "expect" heads 50% of the time. • Flip ten coins and count the number of heads. Do this many times. It is highly likely that in some of the trials, the percentage of heads will be 30% or less. • Flip one hundred coins and count the number of heads. Do this many times. It is highly unlikely that any of the trials will yield a percentage of heads that is 30% or less. • Advanced topic, but related to the above...On TI-83, binomcdf(10,.5,3) = .171875 and binomcdf(100,.5,30) = .00003925. • Basically, if you flip 10 coins, the probability of 3 or fewer heads is about 17%. If you flip 100 coins, the probability of 30 or fewer heads is very close to 0%.

  7. RULES FOR MEANS: • The purpose here is to illustrate important rules (page 418) relating to the mean statistic. • Let Sx= {1,3}. The mean of Sx is 2. • Now, multiply each element by 7, and then add 5, producing the set S7x+5 = {12,26}. • The mean of S7x+5 is 19 = 7(2) + 5. • This illustrates Rule 1 (page 396). In this situation, mean( S7x+5 ) = 7mean(Sx) + 5 • Now, let Sy = {5,9,22}, which has mean =12. • Construct the set Sx+y = {1+5,1+9,1+22,3+5,3+9,3+22} = {6,8,10,12,23,25}. • The mean of Sx+y is 14 = 2 + 12. • This illustrates Rule 2 (page 396). In this situation, mean(Sx+y) = mean(Sx) + mean(Sy).

  8. RULES FOR VARIANCES: • The purpose here is to illustrate important rules (page 420) relating to the variance statistic. • Let Tx = {10,14}. • The mean, standard deviation, and variance of Tx are, respectively, 12, 2, and 4. • Now, multiply each element by 3 and add 5, obtaining the set T3x+5 = {35,47}. • The mean, standard deviation, and variance of T3x+5 are, respectively, 41, 6, and 36. • Note that 36 = 32(4). In other words, var(T3x+5) = 32var(Tx). • Note also that StDev(T3x+5) = 3StDev(Tx). This illustrates Rule 1 (page 400).

  9. RULES FOR VARIANCES: • Now, let Ty = {6,9,12}. The mean, standard deviation, and variance of Ty are, respectively 9, 2.4494897, and 6. • Assume that Tx and Ty are independent sets. • Construct the set Tx-y = {10-6,10-9,10-12,14-6,14-9,14-12} = {-2,1,2,4,5,8}. • The mean, standard deviation, and variance for T are, respectively, 3, 3.16227766, and 10 respectively. • Note that 10 = 4 + 6, or var(Tx-y) = var(Tx) + var(Ty).

  10. RULES FOR VARIANCES: • If you construct the set Tx+y, you will find that • var(Tx+y) = var(Tx-y) = var(Tx) + var(Ty) • This illustrates Rule 2 on page 400. • In a nutshell, with independence, variances add when sets Tx+y and Tx-y are constructed as indicated above. • Note carefully that the standard deviationsdo not add. (Check this out with the examples above.) • You should also note • mean(Tx-y) = 3 = 12 - 9 = mean(Tx) - mean(Ty). • If you construct Tx+y, • you will find that mean(Tx+y) = mean(Tx) + mean(Ty).

  11. Read Pages 392 - 427 Work Ch 7 Teacher Notes RMV Wkst For Section 7.1

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