Mathematical Methods for Physics and Engineering - Matematica.NET

PROBABILITY◮Show that if X and Y are independent random variables then E[XY ]=E[X]E[Y ].LetusconsiderthecasewhereX and Y are continuous random variables. Since X andY are independent f(x, y) =f X (x)f Y (y), so thatE[XY ]=∫ ∞ ∫ ∞xyf X (x)f Y (y) dx dy =∫ ∞xf X (x) dx∫ ∞−∞ −∞−∞−∞An analogous proof exists for the discrete case. ◭yf Y (y) dy = E[X]E[Y ].30.12.2 VariancesThe definitions of the variances of X and Y are analogous to those for thesingle-variable case (30.48), i.e. the variance of X is given by{∑ ∑V [X] =σX 2 = i j (x i − µ X ) 2 f(x i ,y j ) for the discrete case,∫ ∞ ∫ ∞−∞ −∞ (x − µ X) 2 f(x, y) dx dy for the continuous case. (30.132)Equivalent definitions exist for the variance of Y .30.12.3 Covariance and correlationMeans and variances of joint distributions provide useful information abouttheir marginal distributions, but we have not yet given any indication of how tomeasure the relationship between the two random variables. Of course, it maybe that the two random variables are independent, but often this is not so. Forexample, if we measure the heights and weights of a sample of people we wouldnot be surprised to find a tendency for tall people to be heavier than short peopleand vice versa. We will show in this section that two functions, the covarianceand the correlation, can be defined for a bivariate distribution and that these areuseful in characterising the relationship between the two random variables.The covariance of two random variables X and Y is defined byCov[X,Y ]=E[(X − µ X )(Y − µ Y )], (30.133)where µ X and µ Y are the expectation values of X and Y respectively. Clearlyrelated to the covariance is the correlation of the two random variables, definedbyCorr[X,Y ]= Cov[X,Y ] , (30.134)σ X σ Ywhere σ X and σ Y are the standard deviations of X and Y respectively. It can beshown that the correlation function lies between −1 and +1. If the value assumedis negative, X and Y are said to be negatively correlated, if it is positive they aresaid to be positively correlated andifitiszerotheyaresaidtobeuncorrelated.We will now justify the use of these terms.1200