List of top Statistics Questions on Limit Theorems

Let \(X_1,X_2,\ldots\) be a sequence of independent and identically distributed random variables with mean \(4\) and variance \(9\). If \(T_n=\frac{1}{n}\sum_{i=1}^{n}X_i\), then

Let \(X\) be a continuous random variable with mean \(2\) and variance \(4\). Using Chebyshev's inequality, the lower bound for \(P(-4\leq X\leq 8)\) equals (rounded off to two decimal places).

Let \(\{X_n\}_{n\geq 1}\) be a sequence of independent and identically distributed random variables having \(U(0,3)\) distribution. If \(Y_n=\frac{1}{n}\sum_{i=1}^{n}X_i^2,\; n\geq 1\), then \(\{Y_n\}_{n\geq 1}\) converges in probability to (in integer).

Let \(X_1,X_2,\ldots\) be a sequence of independent random variables with