List of top Statistics Questions on Limit Theorems asked in IIT JAM MS

For any real number \( y \), let \( \lfloor y \rfloor \) be the greatest integer less than or equal to \( y \) and let \( \{ y \} = y - \lfloor y \rfloor \). For \( n = 1, 2, \dots \), and for \( x \in \mathbb{R} \), let

\[ f_{2n}(x) = \begin{cases} \frac{\sin x}{x}, & x \neq 0 \\ 1, & x = 0 \end{cases} \quad \text{and} \quad f_{2n-1}(x) = \begin{cases} \frac{\sin x}{x}, & x \neq 0 \\ 1, & x = 0 \end{cases} \]

Then

\[ \lim_{x \to 0} \sum_{k=1}^{100} f_k(x) \text{ equals ............}. \]

Evaluate the limit \[ \lim_{n \to \infty} \left[ n - \frac{n}{e} \left( 1 + \frac{1}{n} \right)^n \right] \text{ equals ............} \]

Evaluate the following limit (round off to 2 decimal places): \[ \lim_{n \to \infty} \frac{\sqrt{n+1} + \sqrt{n+2} + \cdots + \sqrt{n+n}}{\sqrt{n}} \]

Evaluate the limit \[ \lim_{n \to \infty} \frac{1 + \frac{1}{2} + \dots + \frac{1}{n}}{(n + e^n)^{1/n} \log_e n} \]

Let \( \{X_n\}_{n \geq 1} \) be a sequence of independent random variables with \( X_n \) having the probability density function as \[ f_n(x) = \begin{cases} \dfrac{1}{2^{n/2} \Gamma\left( \frac{5}{2} \right)} e^{\tfrac{-x^2}{2}} x^{(\tfrac{n}{2}-1)} , & x > 0, \\ 0, & \text{otherwise}. \end{cases} \] Then \[ \lim_{n \to \infty} \left[ P(X_n > \frac{3n}{4}) + P(X_n > n + 2 \sqrt{2n}) \right] \] equals

Let \( X_1, X_2, \dots, X_n \) be a random sample from the distribution with the probability density function \[ f(x) = \frac{1}{4} e^{-|x-4|} + \frac{1}{4} e^{-|x-6|}, x \in \mathbb{R}. \] Then \( \frac{1}{n} \sum_{i=1}^{n} X_i \) converges in probability to ................

Let \( X_1, X_2, \dots, X_{100} \) be i.i.d. random variables with \( E(X_1) = 0, E(X_1^2) = \sigma^2 \), where \( \sigma>0 \). Let \[ S = \sum_{i=1}^{100} X_i. \, \text{If an approximate value of} \, P(S \leq 30) \, \text{is} \, 0.9332, \, \text{then} \, \sigma^2 \, \text{equals}. \]

If for a suitable \( \alpha>0 \), \[ \lim_{x \to 0} \left( \frac{1}{e^{2x} - 1} - \frac{1}{\alpha x} \right) \] exists and is equal to \( l \) (\( |l|<\infty \)), then \( \alpha = 2, l = -\frac{1}{2} \) is given by