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7.2 Means and Variances of Random Variables

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7.2 Means and Variances of Random Variables

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    1. 7.2 Means and Variances of Random Variables The Mean of a Random Variable

    2. The Mean of a random Variable X An average of the possible values of X, but with an essential change to take into account the fact that not all outcomes need to be equally likely.

    3. Example 7.5 : The Tri-State Pick 3

    4. Mean of a discrete Random Variable Suppose that X is a discrete random variable whose distribution is

    5. Example 7.6 Benford’s Law

    6. Random numbers Benford’s Law

    7. Variance of a discreet Random Variable Suppose that X is a discrete random variable whose distribution is

    8. Example 7.7 Selling Aircraft Parts

    9. Example 7.2 as a table

    10. Statistical Estimation and the Law of large numbers Law of Large Numbers : Draw independent observations at random from any population with finite mean µ. Decide how accurately you would like to estimate µ. As the number of observations drawn increases, the mean x of the observed values eventually approaches the mean µ of the population as closely as you specified and then stays that close.

    11. Example 7.8 Heights of young women

    13. Thinking about the law of large numbers The average results of many independent observations are stable and predictable. Most people believe in an incorrect “law of small numbers” that is we expect even a short sequence of random events to follow the kind of average behavior that appears in the long run. “Streaks” are common in the short run.

    14. Example 7.9: The “Hot Hand” in Basketball People believe that if someone is on a “streak” they are more likely to get the next shot. Statistics show however that shots in basketball behave as if they are independent of each other. Whether or not a player gets the next shot depends on their shooting ability, not on whether or not they are on a streak.

    15. How large is a large number? The more variable the outcomes, the more trials are needed to insure that the mean outcome X is close to the distribution mean µ. For gamblers, the result is unpredictable for the individual because their play is relatively short term. It is not unpredictable for the house in the long run.

    16. Assignment P. 416-417 7.31,7.33

    17. Rules for Means Rule 1: If X is a random variable and a and b are fixed numbers, then µa +bX = a + bµx Rule 2: If X and Y are random variables, then µ X+Y= µx +µy

    18. Example 7.10 Gain Communications

    20. Rule 2 Rule 2 says that the mean of the sum of two variables is the sum of the two means µz = µ2000X + µ3500Y = 10,000,000+1,557,500 =$11,557,500

    21. Rule 1 and 2 combined µz = µ2000x + µ3500Y = 2000µx + 3500µy =(2000)(5000) +(3500)(445) = $11,557,500

    22. Rules for Variances X and Y are independent if knowing that any event involving X alone did or did not occur tells us nothing about the event involving Y. When random variables are not independent , the variation of their sum depends on the correlation between them as well as their individual variances.

    23. The correlation, ?, is a number between -1 and 1 that measures the direction and strength of a linear relationship. The correlation between tow independent random variables is zero. If X is money spent and Y is money saved then Y = 100 –X . This is a perfect linear relationship with a negative slope so ?=-1

    24. Rules For Variances Rule 1: If X is a random variable and a and b are fixed numbers, then s2a+bX = b2sx2 Rule 2: If X and Y are independent random variables, then s2 X+Y = s2x +s2y s2X-Y =s2X + s2Y

    25. Rule 3: Addition Rule for variances of independent of Random variables If X and Y have correlation ?, then s2X+Y =s2X + s2Y + 2?sXs s2X-Y =s2X + s2Y - 2?sXs This is the general addition rule for variances of random variables

    26. Example 7.11 Winning the Lottery The payoff X of a $1 ticket in the Tri-State Pick 3 game is $500 with probability 1/1000 and & $0 the rest of the time. Here is the combined calculation of mean and variance.

    28. Example 7.12: SAT scores SAT Math score X µX = 625 sX = 90 SAT Verbal score Y µX = 590 sY = 100 µX+Y = µX +µY = 625 + 590 =1215 The variance and the standard deviation cannot be computed from the total given. b/c the scores are not independent. Nationally, the correlation for Math and Verbal Scores is about ?= 0.7 s2X + Y = s2X + s2Y + 2?sXsY = (90)2 +(100)2 + (2)(0.7)(90)(100) = 30,700 therefore s= 175

    29. Combining Normal Random Variables Any linear combination of independent normal random variables is also normally distributed. See Example 7.14

    30. 7.14 A round of golf

    31. Assignment P.427-429 7.42, 7.43, 7.44,7.45

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