From the definition of the expected value, the expected value of the product of two random...

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From the definition of the expected value, the expected value of the product of two random variables is E(X+Y)=r₁ r₂ · P(X = T₁, Y = r₂) 72 r1 r2 where the sum is over all possible values of r₁ and r2 that the variable X and Y can take on. (a) Using the definition above formally prove that if the events X = r₁ and Y = r2 are independent (for any r₁ and r2), we have E(XY)= E(X). E(Y). (b) Now, if you have two random variables X and Y and Y = a.X+c, show that if E[X2] + E[X]², then E(XY) E(X). E(Y). (c) If you roll a die 6 times in a row, what is the expected value of the product of the 6 outcomes? From the definition of the expected value, the expected value of the product of two random variables is E(X+Y)=r₁ r₂ · P(X = T₁, Y = r₂) 72 r1 r2 where the sum is over all possible values of r₁ and r2 that the variable X and Y can take on. (a) Using the definition above formally prove that if the events X = r₁ and Y = r2 are independent (for any r₁ and r2), we have E(XY)= E(X). E(Y). (b) Now, if you have two random variables X and Y and Y = a.X+c, show that if E[X2] + E[X]², then E(XY) E(X). E(Y). (c) If you roll a die 6 times in a row, what is the expected value of the product of the 6 outcomes?

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