List of top Statistics Questions asked in IIT JAM MS

Let \( X_1 \) and \( X_2 \) be i.i.d. continuous random variables with the probability density function
\[ f(x) = \begin{cases} 6x(1 - x), & 0 < x < 1 \\ 0, & \text{otherwise} \end{cases} \] Using Chebyshev's inequality, the lower bound of \( P \left( |X_1 + X_2 - 1| \leq \frac{1}{2} \right) \) is
Let \( X \) be a random variable with the moment generating function
\[ M_X(t) = \frac{1}{216} \left( 5 + e^t \right)^3, \quad t \in \mathbb{R}. \] Then \( P(X>1) \) equals
Let \( u, v \in \mathbb{R}^4 \) be such that \( u = \begin{pmatrix} 1 \\ 2 \\ 3 \\ 5 \end{pmatrix} \) and \( v = \begin{pmatrix} 5 \\ 3 \\ 2 \\ 1 \end{pmatrix} \). Then the equation \( u^T x = v \) has
Let \( \{a_n\} \) be a sequence defined as follows:
\[ a_1 = 1 \quad \text{and} \quad a_{n+1} = \frac{7a_n + 11}{21}, \quad n \in \mathbb{N}. \] Which of the following statements is TRUE?