For two independent random variables X and Y,which of the following identities is always true?
A)E(2X + 3Y)= E(X)+ E(Y)+ 5
B)Var(2X + 3Y)=2Var(X)+ 3Var(Y)
C)Var(2X + 3Y)= 4Var(X)+ 9Var(Y)
D)E(2X + 3Y)= 5E(X+Y)

Question

Quizplus · Accepted Answer

DC) The variance of a linear combination of independent random variables is the sum of their variances each multiplied by the square of their coefficients. Thus, $Var(2X + 3Y) = 4Var(X) + 9Var(Y)$.D) The expected value of a linear combination of random variables is the linear combination of their expected values. Therefore, $E(2X + 3Y) = 2E(X) + 3E(Y)$, which can also be expressed as $5E(X) + 5E(Y)$ if $E(X) = E(Y)$, making the statement potentially misleading without this clarification. However, the correct interpretation based on the linearity of expectation is $E(2X + 3Y) = 2E(X) + 3E(Y)$, not directly $5E(X+Y)$, indicating a mistake in my initial assessment. The correct interpretation should focus on the linearity principle directly without implying equal expectations for $X$ and $Y$.

For Two Independent Random Variables X and Y,which of the Following