List of top Statistics Questions on Sampling Distributions asked in GATE ST

Suppose that \(X\) and \(Y\) are independent and identically distributed \(N_p(\mu, \Sigma)\) random vectors, where \(\mu \in \mathbb{R}^p\) and \(\Sigma\) is a positive definite matrix. Let \(\chi^2_m\) denote chi-square distribution with \(m\)-degrees of freedom. Then which of the following statements is correct?

Let \(\{X_n\}_{n\geq1}\) be a sequence of random variables having the following probability mass function \[ P(X_n=x)=\frac{1}{5n}\left(1-\frac{1}{5n}\right)^{x},\quad x=0,1,2,\ldots;\ n\in\mathbb{N}. \] Define \(Z_n=\dfrac{X_n}{n}\), \(n\in\mathbb{N}\), and let \(V\) be a random variable. If \(Z_n\xrightarrow{d}V\), as \(n\to\infty\), then which of the following statements is/are correct?

Let \(\{X_k\}_{k\geq1}\) be a sequence of independent random variables such that \[ X_{2k-1}\sim \text{Bin}(1,\theta),\quad\text{and}\quad X_{2k}\sim \text{Bin}(1,1-\theta),\quad k=1,2,3,\ldots, \] where \(\theta\in(0,1)\). Let \(\{Y_k\}_{k\geq1}\) be another sequence of independent and identically distributed random variables such that \(Y_k\sim\text{Poisson}(\lambda)\), \(\lambda>0\). Define, for \(n\in\mathbb{N}\), \[ S_{2n}=\sum_{k=1}^{n}(X_{2k-1}-X_{2k}+1-2\theta),\quad W_n=\sum_{k=1}^{n}Y_k^2\quad\text{and}\quad \sigma_{2n}^2=2n\theta(1-\theta). \] Then which of the following statements is/are correct?

Let \(U_1,U_2,U_3\) be three independent exponential random variables such that \(U_k\) has the following probability density function
\[ f_k(x)=\begin{cases}ke^{-kx}&\text{if }x>0\\0&\text{otherwise,}\end{cases}\quad k=1,2,3. \]
Let
\[ X=\min\{U_1,U_3\}\quad\text{and}\quad Y=\min\{U_2,U_3\}. \]
Then \(P(X=Y)\) equals ______ (rounded off to two decimal places).