If X is a random variable and $a$, $b$ are real numbers, then which of the following statements are true?

(A) $Var(aX+b) = a^2 Var(X)$

(B) $E(aX+b)= a E(X) + b$

(C) $E(aX+b)= a E(X) - E(b)$

(D) $Var(aX+b)= a Var(X) + b$

Choose the correct answer from the options given below:

For a random variable $X$ and real numbers $a$, $b$, the fundamental properties are:

$E(aX + b) = aE(X) + b$

$Var(aX + b) = a^2 Var(X)$

The expected value is linear, so both multiplication and addition affect it. The variance is affected only by multiplication (squared), since adding a constant shifts all values equally without changing the spread.

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Evaluating statement (A): $Var(aX+b) = a^2 Var(X)$

This matches the variance property exactly.

Statement (A) is true.

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Evaluating statement (B): $E(aX+b) = a E(X) + b$

This matches the expected value property exactly.

Statement (B) is true.

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Evaluating statement (C): $E(aX+b) = a E(X) - E(b)$

Since $b$ is a constant, $E(b) = b$.

This would give $E(aX+b) = a E(X) - b$, which has the wrong sign.

The correct formula has $+b$, not $-b$.

Statement (C) is false.

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Evaluating statement (D): $Var(aX+b) = a Var(X) + b$

The coefficient should be $a^2$, not $a$.

Additionally, adding constant $b$ does not affect variance at all.

Statement (D) is false.

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Only statements (A) and (B) are true.