The journal of f yi, be pi; that Y =22 be Pa etc. The expected value(or defined to be 中1y1+P2y+,十PNyN The variance of y is defined to be 九1(y1-E)2+2(y2-E)2十,十(yx-E) V is the average squared deviation of Y from its expected commonly used measure of dispersion. Other measures of dispersion closely related to v are the standard deviation, o=vv and the co- uppose we have a number r of random variables: R R. if r is a weighted sum (linear combination) of the R R=a1R1十a2R2+.,,+anR then R is also a random variable. (For example Ri, may be the number which turns up on one die; Ro, that of another die, and r the sum of these numbers. In this case n= 2, a1= a2= 1) It will be important for us to know how the expected value and riance of the weighted sum(R)are related to the probability dis- tribution of the ri,..., Ro. We state these relations below; we refer the reader to any standard text for proof.6 The expected value of a weighted sum is the weighted sum of the expected values. I. e, E(R)=a,E(R1)+agE(Ro)+..+ anE(Rn) The variance of a weighted sum is not as simple. To express it we must define "covariance. The covariance of R, and r, is d1=E{[R1-E(R1)]R2-E(R2)] i.e. the expected value of [(the deviation of Ri from its mean) times (the deviation of R2 from its mean). In general we define the covari ance between Ri and r, as σ;=E{[R;-E(R)][R;-ER)] oi may be expressed in terms of the familiar correlation coefficient (p The covariance between R, and R, is equal to [(their correlation) times(the standard deviation of R, times(the standard deviation of m1影:Jpax8 ca Prooa03l4y(New Yor80 The Journal of Finance yl, be pl; that Y = y2 be pz etc. The expected value (or mean) of Y is defined to be The variance of Y is defined to be V is the average squared deviation of Y from its expected value. V is a commonly used measure of dispersion. Other measures of dispersion, closely related to V are the standard deviation, u = .\/V and the co￾efficient of variation, a/E. Suppose we have a number of random variables: R1, . . . ,R,. If R is a weighted sum (linear combination) of the Ri then R is also a random variable. (For example R1, may be the number which turns up on one die; R2, that of another die, and R the sum of these numbers. In this case n = 2, a1 = a2 = 1). It will be important for us to know how the expected value and variance of the weighted sum (R) are related to the probability dis￾tribution of the R1, . . . ,R,. We state these relations below; we refer the reader to any standard text for proof.6 The expected value of a weighted sum is the weighted sum of the expected values. I.e., E(R) = alE(R1) +aZE(R2) + . . . + a,E(R,) The variance of a weighted sum is not as simple. To express it we must define "covariance." The covariance of R1 and Rz is i.e., the expected value of [(the deviation of R1 from its mean) times (the deviation of R2 from its mean)]. In general we define the covari￾ance between Ri and R as ~ij =E ( [Ri-E (Ri) I [Ri-E (Rj)I f uij may be expressed in terms of the familiar correlation coefficient (pij). The covariance between Ri and Rj is equal to [(their correlation) times (the standard deviation of Ri) times (the standard deviation of Rj)l: Uij = PijUiUj 6. E.g.,J. V. Uspensky, Introduction to mathematical Probability (New York: McGraw￾Hill, 1937), chapter 9, pp. 161-81