Question

Let $\{X_n\}_{n \ge 1}$ be a sequence of i.i.d. random variables such that $E(X_i) = 1$ and $\text{Var}(X_i) = 1$. Then the approximate distribution of $\dfrac{1}{\sqrt{n}} \sum_{i=1}^n (X_{2i} - X_{2i-1})$, for large $n$, is
 

• A. $N(0,1)$
• B. $N(0,2)$
• C. $N(0,0.5)$
• D. $N(0,0.25)$

Hint:

For independent variables, the variance of a sum is the sum of variances, enabling direct use of CLT.

Answer 1

The Correct Option is B

Step 1: Construct difference variable.
Let $Y_i = X_{2i} - X_{2i-1}$. Then $E(Y_i) = E(X_{2i}) - E(X_{2i-1}) = 0$, and \[ \text{Var}(Y_i) = \text{Var}(X_{2i}) + \text{Var}(X_{2i-1}) = 1 + 1 = 2. \]

Step 2: Apply Central Limit Theorem (CLT).
By CLT, \[ \frac{1}{\sqrt{n}} \sum_{i=1}^n Y_i \sim N(0, \text{Var}(Y_i)) = N(0,2). \]

Step 3: Conclusion.
Therefore, the approximate distribution is $N(0,2)$.