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Question 32(2 marks) Given that E(aX+b)=aE(X)+b, where E(X) is the expected value of a discrete random variable X and a and b are constants. Prove that Var(aX+b)=a^(2)Var(X)

Question 32(2 32(2 marks)\newlineGiven that E(aX+b)=aE(X)+b E(a X+b)=a E(X)+b , where E(X) E(X) is the expected value of a discrete random variable X X and a a and b b are constants.\newlineProve that Var⁡(aX+b)=a2Var⁡(X) \operatorname{Var}(a X+b)=a^{2} \operatorname{Var}(X)

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Q. Question 32(2 32(2 marks)\newlineGiven that E(aX+b)=aE(X)+b E(a X+b)=a E(X)+b , where E(X) E(X) is the expected value of a discrete random variable X X and a a and b b are constants.\newlineProve that Var⁡(aX+b)=a2Var⁡(X) \operatorname{Var}(a X+b)=a^{2} \operatorname{Var}(X)
  1. Define Variance: Question Prompt: Prove that Var(aX+b)=a2Var(X)\text{Var}(aX+b)=a^{2}\text{Var}(X) for a discrete random variable XX and constants aa and bb.
  2. Calculate E(aX+b)E(aX+b): Step 11: Recall the definition of variance, Var(Y)=E(Y2)−[E(Y)]2\text{Var}(Y) = E(Y^2) - [E(Y)]^2. Apply this to Y=aX+bY = aX + b.
  3. Expand (aX+b)2(aX+b)^2: Step 22: Calculate E(aX+b)E(aX+b). Using the linearity of expectation, E(aX+b)=aE(X)+bE(aX+b) = aE(X) + b.
  4. Apply Linearity of Expectation: Step 33: Calculate E((aX+b)2)E((aX+b)^2). Expand the square, E((aX+b)2)=E(a2X2+2abX+b2)E((aX+b)^2) = E(a^2X^2 + 2abX + b^2).
  5. Substitute into Variance Formula: Step 44: Apply the linearity of expectation to each term, E(a2X2+2abX+b2)=a2E(X2)+2abE(X)+b2E(a^2X^2 + 2abX + b^2) = a^2E(X^2) + 2abE(X) + b^2.
  6. Expand Square: Step 55: Substitute E(aX+b)E(aX+b) and E((aX+b)2)E((aX+b)^2) into the variance formula. Var(aX+b)=[a2E(X2)+2abE(X)+b2]−[aE(X)+b]2.\text{Var}(aX+b) = [a^2E(X^2) + 2abE(X) + b^2] - [aE(X) + b]^2.
  7. Simplify Variance Expression: Step 66: Expand the square in the variance formula. [aE(X)+b]2=a2[E(X)]2+2abE(X)+b2[aE(X) + b]^2 = a^2[E(X)]^2 + 2abE(X) + b^2.
  8. Cancel Out Terms: Step 77: Simplify the variance expression. Var(aX+b)=a2E(X2)+2abE(X)+b2−(a2[E(X)]2+2abE(X)+b2)\text{Var}(aX+b) = a^2\text{E}(X^2) + 2ab\text{E}(X) + b^2 - (a^2[\text{E}(X)]^2 + 2ab\text{E}(X) + b^2).
  9. Recognize Pattern: Step 88: Cancel out terms. Var(aX+b)=a2E(X2)−a2[E(X)]2\text{Var}(aX+b) = a^2\text{E}(X^2) - a^2[\text{E}(X)]^2.
  10. Recall Variance Definition: Step 99: Recognize that a2E(X2)−a2[E(X)]2=a2(E(X2)−[E(X)]2)a^2E(X^2) - a^2[E(X)]^2 = a^2(E(X^2) - [E(X)]^2).
  11. Recall Variance Definition: Step 99: Recognize that a2E(X2)−a2[E(X)]2=a2(E(X2)−[E(X)]2)a^2E(X^2) - a^2[E(X)]^2 = a^2(E(X^2) - [E(X)]^2). Step 1010: Recall the definition of variance for XX, Var(X)=E(X2)−[E(X)]2\text{Var}(X) = E(X^2) - [E(X)]^2. Substitute to find Var(aX+b)=a2Var(X)\text{Var}(aX+b) = a^2\text{Var}(X).

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