Suppose $X_1, X_2, \dots$ are independent random variables. Assume that $X_1, X_3, \dots$ are identically distributed with mean $\mu_1$ and variance $\sigma_1^2$, while $X_2, X_4, \dots$ are identically distributed with mean $\mu_2$ variance $\sigma_2^2$. Let $S_n = X_1 + X_2 + \dots + X_n$. Then $\frac{S_n - a_n}{b_n}$ converges in distribution to $N(0,1)$ if
$a_n = \frac{n(\mu_1 + \mu_2)}{2}$ and $b_n = \sqrt{n} \sqrt{\frac{\sigma_1^2 + \sigma_2^2}{2}}$
The question asks for the conditions ($a_n$ and $b_n$) under which the standardized sum of independent random variables converges in distribution to a standard normal distribution, $N(0,1)$. The random variables are split into two groups: odd-indexed ($X_1, X_3, \dots$) with mean $\mu_1$ and variance $\sigma_1^2$, and even-indexed ($X_2, X_4, \dots$) with mean $\mu_2$ and variance $\sigma_2^2$. We are given the sum $S_n = X_1 + X_2 + \dots + X_n$.
Convergence Conditions via Central Limit Theorem
For the standardized sum $\frac{S_n - a_n}{b_n}$ to converge in distribution to $N(0,1)$, $a_n$ typically represents the expected value of $S_n$ (i.e., $E[S_n]$) and $b_n$ is typically related to the standard deviation of $S_n$ (i.e., $\sqrt{Var(S_n)}$). This is a consequence of the Central Limit Theorem (CLT), particularly variants like the Lindeberg-Feller CLT for sums of independent, non-identically distributed random variables.
Calculating Expected Value $E[S_n]$
Let's consider the expected value of $S_n$. The number of odd-indexed terms up to $n$ is $\lceil n/2 \rceil$, and the number of even-indexed terms is $\lfloor n/2 \rfloor$.
For large $n$, both counts are approximately $n/2$.
The expected value $E[S_n]$ can be approximated as:
$E[S_n] \approx \frac{n}{2} \mu_1 + \frac{n}{2} \mu_2 = \frac{n(\mu_1 + \mu_2)}{2}$
Thus, we expect $a_n = \frac{n(\mu_1 + \mu_2)}{2}$.
Calculating Variance $Var(S_n)$
Similarly, the variance of $S_n$, due to independence, can be approximated:
$Var(S_n) = \sum_{i=1}^n Var(X_i)$
For large $n$, this is approximately:
$Var(S_n) \approx \frac{n}{2} \sigma_1^2 + \frac{n}{2} \sigma_2^2 = \frac{n(\sigma_1^2 + \sigma_2^2)}{2}$
For convergence to $N(0,1)$, we need the scaling factor $b_n$ such that $\frac{S_n - E[S_n]}{b_n}$ converges. Typically, $b_n = \sqrt{Var(S_n)}$. Therefore:
$b_n = \sqrt{\frac{n(\sigma_1^2 + \sigma_2^2)}{2}} = \sqrt{n} \sqrt{\frac{\sigma_1^2 + \sigma_2^2}{2}}$
Matching with Options
Comparing our derived expressions for $a_n$ and $b_n$ with the given options:
- Option 1: $a_n = \frac{n(\mu_1 + \mu_2)}{2}$ and $b_n = \sqrt{n} \sqrt{\frac{\sigma_1^2 + \sigma_2^2}{2}}$
- Option 2: $a_n = \frac{n(\mu_1 + \mu_2)}{2}$ and $b_n = \frac{n(\sigma_1 + \sigma_2)}{2}$
- Option 3: $a_n = n(\mu_1 + \mu_2)$ and $b_n = \sqrt{n} \frac{(\sigma_1 + \sigma_2)}{2}$
- Option 4: $a_n = n(\mu_1 + \mu_2)$ and $b_n = \sqrt{n} \sqrt{\frac{\sigma_1^2 + \sigma_2^2}{2}}$
Our derived values match Option 1 exactly. This option uses the average mean and the average variance, scaled appropriately by $n$ and $\sqrt{n}$ respectively, consistent with the requirements for the CLT.
Important Questions from Central Limit Theorems
- Let $X_1, X_2, . . .$ be a sequence of independent and identically distributed random variables with $E(X_1) = 0, E(X_1^2) = 1, E(X_1^3) = 0, E(X_1^4) = 3$. Let $S_n = \sum_{i=1}^n X_i, T_n = \sum_{i=1}^n X_i^2, U_n = \sum_{i=1}^n X_i^3$ and $V_n = \sum_{i=1}^n X_i^4$. Then, which of the following statements are true?
- Let $\{X_i; i \ge 1\}$ be a sequence of independent random variables each having a normal distribution with mean 2 and variance 5. Then which of the following are true
- For $n \ge 1$, let $X_n$ be a Poisson random variable with mean $n^2$. Which of the following are equal to $\frac{1}{\sqrt{2\pi}} \int_2^\infty e^{-x^2/2} dx$?
Let $\{X_n : n \ge 1\}$ be a sequence of independent and identically distributed random variables and the probability mass function of $X_1$ is the following;
$P(X_1 = 1) = P(X_1 = 3) = \frac{1}{2}.$ If $Y_n = X_1 + \cdots + X_n$,
then which of the following statements are correct?
- Let $X_i$'s be independent random variables such that $X_i$'s are symmetric about 0 and $\text{Var}(X_i) = 2i-1$, for $i \ge 1$. Then,
$$\lim_{n\to\infty} P(X_1 + X_2 + \cdots + X_n > n \log n)$$
