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Let X and Y be two independent random variables. Which one of the relations between expectation (E), variance (Var) and covariance (Cov) given below is FALSE?

  1. E(XY) = E(X) E(Y)
  2. Cov(X, Y) = 0
  3. Var (X + Y) = Var (X) + Var(Y)
  4. E(X2Y2) = (E(X))2 (E(Y))2

Answer (Detailed Solution Below)

Option 4 : E(X2Y2) = (E(X))2 (E(Y))2

Detailed Solution

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Explanation:

For X and Y be two independent random variables.

i) E(XY) = E(X) E(Y)

ii) Cov (X,Y) = E (XY) - E(X) E(Y) ⇒ E(X) E(Y) - E(X) E(Y) = 0

iii) Var (X + Y) = Var(X) + Var(Y)

iv) E(X2Y2) = E(X2) E(Y2)

∴ E(X2Y2) = E(X2) ⋅ E(Y2) is the incorrect option

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