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6-18). QED The correlation coefficient of Xi and X2 is defined by cov{X X } corriX^XJ = —. 6-18), which evidently implies that - 1 < corriXpX^ < + 1 . 6-20) Note that it is possible for either equality in Eq. 6-20) to hold. 6-19) reduces to + 1 ; we say that Xi is "maximally correlated" with with itself. On the other hand, if X2 = — Xi we have cov{XvX2} = coviX^-X^ = ( X ^ - X ^ ) (X^-Xy) = - v a r t X J = -(sdeviXJ) 2 , 34 1. " We say that two random variables X\ and X are "uncorrelated" if and only if cov{Xi,X2} = 0.

Distribution function F(x), then If the random variable X has F\X) = U. 6-10) where U is the unit uniform random variable, U(0,1). Proof. Since the distribution function F rises monotonically from 0 at x= — » to 1 at x= + 0 ° , then the transformation y = F{x) maps the interval — q o < X < q o onto the interval 0 < y < l ; this means that the random variable Y=F\X) can have sample values only inside the unit interval. [0,l]. 2-9b) between the density function and the distribution function. Thus, the density function of Y is 0 outside the unit interval and 1 inside the unit interval, which means that y must be the random variable U(0,1).

Eq. 2-1)], INTEGER VALUES. PIN) s Prob{X=/i}. 7-1) We again call Ρ the DENSITY FUNCTION of X, and Eq. 7-1) may be regarded as its definition. Using the addition law of probability, we easily deduce that the probability of finding a value for X between N\ and N inclusively is [cf. Eq. 2-2)J 2 n 2 Prob{X€[n ,,i J}= 1 2 Σ 1 ^ t 7 - 2 " ) η— In particular, since X will surely take some integer value between — oo and + 0 0 , then [cf. Eq. 2-3)1 40 1. 7-3) -co an equation that is called the "normalization condition" for P.