JAM Mathematical Statistics (MS) - Previous Year Papers Previous Year Questions

An eigenvector of the matrix \( M = \begin{pmatrix} 2 & 1 & 0 \\ 0 & 2 & 1 \\ 0 & 0 & 2 \end{pmatrix} \) is

Let \(E\) and \(F\) be two events with \(P(E)=0.7, P(F)=0.4\) and \(P(E\cap F^c)=0.4\). Then \(P(F|E\cup F^c)\) is equal to

The equation \[3x^2-12x+11+\frac{1}{5}\left(x^3-6x^2+11x-6\right)=0\] has

Let \(X_1,\ldots,X_n\) be a random sample from a population with probability density function \[f_\theta(x)=\begin{cases}\theta e^{-\theta x}, & x>0,\\ 0, & \text{otherwise},\end{cases}\] where \(\theta>0\) is an unknown parameter. Then, the uniformly minimum variance unbiased estimator for \(\frac{1}{\theta}\) is Suppose \(\{a_n\}, \{b_n\}\) are sequences such that \( a_n > 0, b_n > 0 \) for all \( n \ge 1 \). Given that \( \sum a_n \) converges and \( \sum b_n \) diverges, which of the following statements is (are) necessarily FALSE? Let \(X\) and \(Y\) be independent exponentially distributed random variables with means \(1/4\) and \(1/6\) respectively. Let \(Z=\min\{X,Y\}\). Then \(E(Z)=\)___________.

Let \[ P = \begin{bmatrix} 1 & 2 & 0 & 2 \\ -1 & -2 & 1 & 1 \\ 1 & 2 & -3 & -7 \\ 1 & 2 & -2 & -4 \end{bmatrix}. \] Then rank of \(P\) equals

The imaginary parts of the eigenvalues of the matrix \[P = \begin{pmatrix}3 & 2 & 5\\2 & -3 & 6\\0 & 0 & -3\end{pmatrix}\] are