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P(X = xY = y) = X x P{Y=y}(X = x) = 1, using the subscript notation P{Y=y} of Section 23 52 We can also find the expectation and variance of X with respect to this conditional distribution That is, if we know that the value of Y is fixed at y, then 3 = 3 =) = " # = " # = = =.
P xy ognex. The result P ( Y ≤ 075 X = 05 ) = 5/6, mentioned above, is geometrically evident in the following sense The points (x,y,z) of the sphere x 2 y 2 z 2 = 1, satisfying the condition x = 05, are a circle y 2 z 2 = 075 of radius on the plane x = 05 The inequality y ≤ 075 holds on an arc The length of the arc is 5/6 of the length. 0, X and Y are said to be negatively correlated When Cov(X,Y) = 0, X and Y are said to be uncorrelated, and in general this is weaker than independence of X and Y there are examples of uncorrelated rvs that are not independent Note in passing that Cov(X,X) = Var(X) The correlation coefficient of X,Y is defined by ρ = σ X,Y σ Xσ Y,.
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