208 CHAPTER 5 Discrete Probability Distributions Parameters of a Probability Distribution Remember that with a probability distribution, we have a description of a population instead of a sample, so the values of the mean, standard deviation, and variance are parameters, not statistics. The mean, variance, and standard deviation of a discrete probability distribution can be found with the following formulas: FORMULA 5-2 VarianceS 2 for a probability distribution s 2 = Σ31x - m2 2 # P1x24 (This format is easier to understand.) FORMULA 5-1 MeanM for a probability distribution m = Σ3x # P1x24 FORMULA 5-3 VarianceS 2 for a probability distribution s 2 = Σ3x2 # P1x24 - m 2 (This format is easier for manual calculations.) FORMULA 5-4 Standard deviationS for a probability distribution s = 2Σ3x 2 # P1x24 - m 2 Round-Off Rule for M, S, and S 2 from a Probability Distribution Round results by carrying one more decimal place than the number of decimal places used for the random variable x. If the values of x are integers, round m, s, and s 2 to one decimal place. When applying Formulas 5-1 through 5-4, use the following rule for rounding results. Exceptions to Round-Off Rule In some special cases, the above round-off rule results in values that are misleading or inappropriate. For example, with four-engine jets the mean number of jet engines working successfully throughout a flight is 3.999714286, which becomes 4.0 when rounded, but that is misleading because it suggests that all jet engines always work successfully. Here we need more precision to correctly reflect the true mean, such as the precision in 3.999714. Expected Value The mean of a discrete random variable x is the theoretical mean outcome for infinitely many trials. We can think of that mean as the expected value in the sense that it is the average value that we would expect to get if the trials could continue indefinitely. DEFINITION The expected value of a discrete random variable x is denoted by E, and it is the mean value of the outcomes, so E = m and E can also be found by evaluating E = Σ3x # P1x24, as in Formula 5-1. P R in p Jerry and Marge Beat the Lottery Jerry and Marge Selbee made $26 million in lottery winnings, and it wasn’t due to an abundance of good luck. Jerry saw a brochure for a Michigan lottery game called Winfall and quickly realized that because the jackpot kept building, the expected value of a ticket actually became positive once the jackpot reached $5 million. With this jackpot size, it became profitable to buy a large number of lottery tickets. Initially, Jerry and Marge invested $515,000 in Winfall tickets and got back $853,000 in winnings. They then moved on to play the similar Cash WinFall game in Massachusetts. In total, they spent $18 million on losing tickets in the process of winning their $26 million. And it was all legal—they just did the math! J M m m w it t d f d l k
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