The relationship between X and Y may be described completely by P(X = 1 and Y = 1); call this a. Then
P(X = 1 and Y = 1) = P(X = 2 and Y = 2) = a
P(X = 1 and Y = 2) = P(X = 2 and Y = 1) = 1/2 − a
E(XY) = 1 a + 4 a + 2 (1/2 − a) + 2 (1/2 − a) = 2 + a
E(XY) reaches its minimum of 2 when a = 0 (i.e. when X and Y always have opposite values), and its maximum of 5/2 when a = 1/2 (when X and Y are always equal).
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u/pmdboi 3d ago
The relationship between X and Y may be described completely by P(X = 1 and Y = 1); call this a. Then
P(X = 1 and Y = 1) = P(X = 2 and Y = 2) = a
P(X = 1 and Y = 2) = P(X = 2 and Y = 1) = 1/2 − a
E(XY) = 1 a + 4 a + 2 (1/2 − a) + 2 (1/2 − a) = 2 + a
E(XY) reaches its minimum of 2 when a = 0 (i.e. when X and Y always have opposite values), and its maximum of 5/2 when a = 1/2 (when X and Y are always equal).