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CHAPTER 7 Section 7.2 – Means and Variances of Random Variables MEANS AND VARIANCES OF RANDOM VARIABLES Probability is the mathematical language that describes the long-run regular behavior of random phenomena. The probability distribution of a random variable is an idealized relative frequency distribution. EXAMPLE 7.5 - THE TRI-STATE PICK 3 See example 7.5 on p.407 Probability distribution of X: Payoff X: Probability: $0 0.999 $500 0.001 Mean of the random variable X is found by: $500 1 999 + $0 = $0.50 1000 1000 In other words, you are expected to lose $0.50 per ticket if many tickets are purchased over time. THE MEAN OF A RANDOM VARIABLE The mean, , of a set of observations is their ordinary average. The mean of a random variable X is also 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. MEAN 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 To find the mean of X, multiply each possible value by its probability, then add all products; = x1 p1 + x2 p2+ …….xk pk = x p MEAN AND EXPECTED VALUE The mean of a probability distribution describes the long-run average outcome. You will often find the mean of a random variable X called expected value of X. The common symbol μ, the Greek letter mu, is used to represent the mean of a probability distribution (expected value). Some other common notations include: (this is the most common) () THE VARIANCE OF RANDOM VARIABLE The mean is a measure of the center of a distribution. The variance and the standard deviation are the measures of spread that accompany the choice of the mean to measure center. Recall from chapter 2 that the variance of a data set is written as 2 and it represents an average of the squared deviation from the mean. To distinguish between the variance of a data set and the variance of a random variable X, we write the variance of a random variable X as THE VARIANCE OF RANDOM VARIABLE Definition: Suppose that X is a discrete random variable whose probability distribution is Value: x1 x2 x3 … Probability: p1 p2 p3 … and that µX is the mean of X. The variance of X is Var(X) X2 (x1 X ) 2 p1 (x 2 X ) 2 p2 (x 3 X ) 2 p3 ... (x i X ) 2 pi The standard deviation σx of X is the square root of the variance. EXAMPLE 7.7 - SELLING AIRCRAFT PARTS Gain Communications sells aircraft communications units to both the military and the civilian markets. Next year’s sales depend on market conditions that cannot be predicted exactly. Gain follows the modern practice of using probability estimates of sales. The military division estimates its sales as follows: Units sold: Probability: 1000 0.1 3000 0.3 5000 0.4 Calculate the mean and variance of X See p.411 to check your answers 10,000 0.2 STATISTICAL ESTIMATION AND THE LAW OF LARGE NUMBERS To estimate μ, we choose a SRS of young women and use the sample mean to estimate the unknown population mean μ. Statistics obtained from probability samples are random variables because their values would vary in repeated samplings. It seems reasonable to use to estimate μ. A SRS should fairly represent the population, so the mean of the sample should be somewhere near the mean μ of the population. Of course, we don’t expect to be exactly equal to μ, and realize that if we choose another SRS, the luck of the draw will probably produce a different . LAW OF LARGE NUMBERS If we keep on adding observations to our random sample, the statistic is guaranteed to get as close as we wish to the parameter μ and then stay that close. This remarkable fact is called the law of large numbers. The law of large numbers states the following: 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 of the observed values eventually approaches the mean μ of the population as closely as you specified and then stays that close. See example 7.8 on p.414 THE “LAW OF SMALL NUMBERS” Both the rules of probability and the law of large numbers describe the regular behavior of chance phenomena in the long run. Psychologists have discovered most people believe in an incorrect “law of small numbers” That is, we expect even short sequences of random events to show the kind of average behavior that in fact appears only in the long run. HOW LARGE IS A LARGE NUMBER? The law of large numbers says that the actual mean outcome of many trials gets close to the distribution mean μ as more trials are made. It doesn’t say how many trials are needed to guarantee a mean outcome close to μ. Homework: p.412-417 #’s 24, 29, & 32 Work on ch.6 & 7 extra credits