If X and Y are two random variables then v(x+y), when x is independent of y

Option a: v(x) + v(y). Solution: Concept: Variance of Sum of Two Random Variables In general: V(X + Y) = V(X) + V(Y) + 2text{Cov}(X, Y) When X and Y are independent: text{Cov}(X, Y) = 0 therefore V(X + Y) = V(X) + V(Y) + 2(0) = V(X) + V(Y) Note: Option C includes +2V(X,Y) which would apply when they are NOT independent. Option B uses subtraction (which applies to V(X-Y) for independent variables). For independent X and Y: V(X+Y) = V(X) + V(Y). Hence, Option A is the correct answer.

If $X$ and $Y$ are two random variables then $v(x+y)$, when $x$ is independent of $y$

The correct answer is Option a: $v(x) + v(y)$. **Concept: Variance of Sum of Two Random Variables** In general: $$V(X + Y) = V(X) + V(Y) + 2\text{Cov}(X, Y)$$ **When $X$ and $Y$ are independent:** $$\text{Cov}(X, Y) = 0$$ $$\therefore V(X + Y) = V(X) + V(Y) + 2(0) = V(X) + V(Y)$$ Note: Option C includes $+2V(X,Y)$ which would apply when they are NOT independent. Option B uses subtraction (which applies to $V(X-Y)$ for independent variables). For independent $X$ and $Y$: $V(X+Y) = V(X) + V(Y)$. Hence, **Option A** is the correct answer.

Central Tendency & DispersionMTP March 21Question 3167 of 473

All Questions

If $\displaystyle X$ and $\displaystyle Y$ are two random variables then $\displaystyle v(x+y)$ , when $\displaystyle x$ is independent of $\displaystyle y$

Options

\displaystyle v(x) + v(y)

\displaystyle v(x) + v(y) - 2v(x,y)

\displaystyle v(x) + v(y) + 2v(x,y)

\displaystyle v(x) - v(y)

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Correct Answer

✅ Option a — $\displaystyle v(x) + v(y)$

All Options:

A $\displaystyle v(x) + v(y)$
B $\displaystyle v(x) + v(y) - 2v(x,y)$
C $\displaystyle v(x) + v(y) + 2v(x,y)$
D $\displaystyle v(x) - v(y)$

Detailed Solution & Explanation

**Concept: Variance of Sum of Two Random Variables** In general:

V(X + Y) = V(X) + V(Y) + 2\text{Cov}(X, Y)

**When

\displaystyle X

and

\displaystyle Y

are independent:**

\text{Cov}(X, Y) = 0

\therefore V(X + Y) = V(X) + V(Y) + 2(0) = V(X) + V(Y)

Note: Option C includes

\displaystyle +2V(X,Y)

which would apply when they are NOT independent. Option B uses subtraction (which applies to

\displaystyle V(X-Y)

for independent variables). For independent

\displaystyle X

and

\displaystyle Y

\displaystyle V(X+Y) = V(X) + V(Y)

. Hence, **Option A** is the correct answer.

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If X\displaystyle XX and Y\displaystyle YY are two random variables then v(x+y)\displaystyle v(x+y)v(x+y), when x\displaystyle xx is independent of y\displaystyle yy