251var2 4/19/06 (Open in page layout)Roger Even Bove FORMULAS FOR FUNCTIONS OF RANDOM VARIABLESI. Basic Computational Formulas for Descriptive StatisticsConsider the following set of observations: Observation number x y 1 7 3 2 15 6 3 2 9 You can easily verify that x = 8 and y = 6. The formula for the sample variance is 1222nxnxsxSo that 4321922781383215712222222nxnxsxand 43xs .Similarly 92108126136396312222222nynysy and3ysThe formula for the sample covariance is yxnxynnyxnxysxy111So that , for the numbers above 5.714412921)6)(8(3)9)(2()6)(15()3)(7(131xyspg. 58The only thing that we can usually learn from a covariance is whether the variables x and y move together or in opposite directions. If a covariance is positive the two variables tend to move together. If the covariance is negative the two variables tend to move in the same direction. To find out about the strength of the relationship we compute the correlation. The correlation can only have values between 1.0 and -1.0. The sign of the correlation means the same thing as the sign of the covariance. A correlation close to 1.0 is referred to as a strong positive correlation. A correlation close to -1.0 is a strongnegative correlation. If the correlation is 1.0, x and y will tend to move proportionally, that is whenx rises, y will rise and when x falls y will fall. When x takes a big jump, y will take a big jump. When x takes a small jump, y takes a small jump. If the correlation is -1 we have the same proportionality, but now if x jumps, y will jump in the opposite direction. If the correlation is zero or close to zero, it is weak, which means there is not much tendency of y to do anything in particular ifx moves. The formula for sample correlation is yxxyxysssr and we know that 5.7xys,43xs and 3ys, so that, for the numbers above .381.03435.7xyrThe negative covariance tells us that x and y have a tendency to move in opposite directions. The negative correlation tells us the same thing, but the fact that it is closer to zero than 1 leads us to feel that the correlation is weak. Actually statisticians tend to measure strength on a zero to one scale by squaring the correlation. In this case 145.0381.022xyr, which appears quite weak, though far from nonexistent. The sample covariance is regarded as an estimate of the true or population covariance, just as the sample correlation is regarded as an estimate of the population correlation. Formulas for computing these from a population all of whose points are known are not given here. The next section will deal with computing population covariances and correlations when probabilities are known.pg. 592II. FORMULAS FROM PROBABILITYLet the following table describe the joint probabilities of x and y: y sum x 7 15 2 xpx xp xxp xxp 3 0.1 0.2 0.0 0.3 3 0.3 0.9 2.7 6 0.1 0.0 0.3 0.4 6 0.4 2.9 14.4 9 0.0 0.10.2 0.3 9 0.3 2.7 24.3 yp0.2 0.3 0.5 1.0 1.0 6.0 xE 41.4 2xE yyp1.4 4.5 1.0 6.9 yE ypy29.8 67.5 2.0 79.3 2yENote that xxpxEx 6.0 and 222xxxE = 4.564.412and similarly yypyEy= 6.9 and 222yyyE = 69.319.63.792This implies that 32.24.5 x and that 63.569.31 y.The formula for the covariance is yxyxxyyxpxyxyE,.We call this a population covariance, since the probabilities presumably refer to all values of x and y. The most difficult part of this formula is the evaluation of the expected value of xy, xyE. The idea here is to multiply each possible pair of values of x and y by the joint probability of the pair. One way to do this is to take the joint probability table and to add the values of x and y to it. Notice that in the table below the probabilities (like 0.1) are in exactly the same place as in the joint probability table above and are followed by the corresponding x and y. 0.36)2)(9(2.0)15)(9(1.0)7)(9(0.0)2)(6(3.0)15)(6(0.0)7)(6(1.0)2)(3(0.0)15)(3(2.0)7)(3(1.0,yxpxyxyE So, 4.5)9.6)(0.6(0.36 yxxyxyE . Once again we find a negative covariance, indicating a tendency of x and y to move in opposite directions. To measure the strength of the relationship, we must compute the correlation. As with the sample correlation, the population correlation is computed by dividing the covariance by the standard deviations of x and y. This time the formula for the correlation reads: yxxyxy . From above, we know that 32.2,4.5 xxy and 63.5y . Thus .41.0)63.5)(32.2(4.5xyAs with the sample correlation this can only take values between negative and positive one. Since, if we square -0.41 we get 0.17, this too is a weak correlation.pg. 603In many situations, especially with population correlations, we are likely to need the covariance and know the correlation. The formula for population correlation can be rewritten as yxxyxy .Thus, if we know 32.2,41.0 xxy and 63.5y , we can compute the covariance.4.5)63.5)(32.2(41.0 xy The corresponding formula for the sample covariance is yxxyxyssrs .pg. 614III FUNCTIONS OF RANDOM VARIABLESA. Functions of a Single Random Variable. 1. The Mean. If we know the mean of the distribution of a random variable, we can easily find the mean of a linear function of the same random variable. For example if we know the mean of x we can find the mean of 75 x. In the following let a and bbe constants that either multiply x or are added tox. Of course, )(xxpxEx, but these formulas apply to x, the sample mean, as well. a) If b is a constant, then bbE . For example 77 E.
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